Film thickness controller

ABSTRACT

There is disclosed a film thickness controller which contains a large dead time and in which, in order to solve a problem due to the dead time, past data is stored in a memory and a state variable is calculated on the basis of the stored data to control a system. The film thickness controller includes an adjusting mechanism including a plurality of die lips having interference effect. In order to solve a problem due to the interference effect, the film thickness controller combines a plurality of basic control systems to control the adjusting mechanism including the plurality of die lips.

This is a division of application Ser. No. 311,223 filed Feb. 15, 1989.

FIELD OF THE INVENTION AND RELATED ART STATEMENT

The present invention relates to a film thickness controller for use inan extrusion molding apparatus and a flowing type molding apparatus suchas a film or sheet manufacturing apparatus.

A conventional film thickness controller is now described briefly.

An extrusion molding apparatus for manufacturing film or sheet isrequired to manufacture a molded product such as film or sheet havingthickness maintained to a predetermined value. An example of aconventional apparatus having a die provided with an adjusting mechanismwhich can adjust thickness of film along the width thereof is nowdescribed with reference to FIGS. 28 to 30. Molten plastic is fed froman extruder 1b (FIG. 28) to a die 2b. The molten plastic is expanded ina manifold 3b in the width direction perpendicular to paper of FIG. 28showing the die 2b and flows down from a slit-shaped outlet 5b of dielips 4b. Then, the molten plastic flowing down from the outlet 5b iscooled by a cooling roller 6b and solidified to be film 7b so that thefilm is wound on a winder 10b.

A thickness gauge 11b measures thickness of the film 7b. The thicknessgauge 11b utilizes radiation due to the natural disintegration ofradioactive substance to measure thickness of the film 7b in accordancewith degree of reduction of the radiation intensity when the radiationpasses through the film 7b. The thickness gauge includes a singledetection element which is moved in the reciprocating manner along thewidth of the film 7b to measure thickness of the film 7b along thewidth.

It is required that the thickness of the film 7b is maintained to be apredetermined thickness along the width. However, since it is difficultthat molten plastic passes through a narrow gap of the die 2b in thesame speed along the width, the thickness of the film 7b is notnecessarily identical along the width thereof.

Accordingly, thickness adjusting mechanisms 12b which serve to change adischarge amount of molten plastic along the length of the slot of thedie lips 4b are disposed dispersedly along the length of the slot of thedie lips 4b. As the thickness adjusting mechanism 12b, there are thefollowing types, for example:

(1) Heater type: A multiplicity of heaters are embedded in the die lips4b along the length of the slot of the die lips 4b and are controlled tochange a temperature generated therefrom so that the viscosity of themolten plastic therein is changed and the flowing speed of the moltenplastic is changed to control the discharge amount of the moltenplastic.

(2) Bolt type: A multiplicity of screws are disposed along the length ofthe slot of the die lips 4b to change a gap space of the dischargeoutlet 5b of the slot of the die lips 4b mechanically or thermally orelectrically so that the discharge amount of the molten plastic iscontrolled.

Accordingly, the thickness of the film 7b can be automaticallycontrolled by adjustment of the thickness adjusting mechanism 12b.

For example, as shown in FIGS. 2 and 3, a multiplicity of heaters 12aare embedded in a die 2a at both sides of a gap 3a and the heaters 12aare distributed along the width so that the speed of molten plasticflowing through the gap 3a is maintained to constant.

At this time, when a temperature of the heater 12a which is located in aplace where thickness of film 6a is thick is reduced, a temperature ofmolten plastic being in contact with the die 2a is reduced and theviscosity of the molten plastic is increased. Accordingly, the flowingspeed of the molten plastic therein is reduced. Thus, the thickness of aportion of the film 6a corresponding to the place where the temperatureof the heater 12a is reduced is reduced so that the thicker portion ofthe film 6a is corrected. Conversely, when the thickness of the film 6ais small, the temperature of the heater 12 which is disposed in a placewhere the thickness of film is small is increased so that the speed ofthe molten plastic flowing through the place is increased and thethickness of the film 6a therein is increased to correct the thicknessof film.

FIG. 4 is a block diagram of a conventional thickness controller. When adifference between a film thickness measured by a thickness gauge 10 anda set value for the film thickness is applied to a controller 13, thecontroller 13 supplies a command to a heater 12a to change a temperatureof heat generated by the heater 12a. When the temperature of the heater12a is changed, the flowing speed of the molten plastic in the die 2a ischanged so that thickness of a portion of the film corresponding to theplace where the temperature of the heater is changed can be controlled.

FIG. 30 is a block diagram of a conventional thickness controller forone operating terminal device of the thickness adjusting mechanism 12b.A controller 13b is supplied with a difference between a thickness b ofa portion of the film measured by the thickness gauge 11b and a setvalue a of thickness. The controller 13b calculates an amount ofoperation for the adjusting mechanism 12b corresponding to the portionof the film measured by the thickness gauge 11b and supplies it to theadjusting mechanism 12b. When the mechanism 12b is operated, a dischargeamount of molten plastic in the die lips 4b is changed and thickness ofthe portion of the film controlled by the mechanism 12b is changed toeffect the thickness control. The thickness control over the whole widthof the film can be made by provision of the number of the control loopblocks of FIG. 30 corresponding to the number of places in which thethickness control is performed.

The conventional film thickness controller as described above hasdrawbacks as follows:

(1) There is a dead time L₁ due to movement of the film from the outletof the die to the thickness gauge 10 until variation of thickness of thefilm is detected by the thickness gauge 10 after the variation has beenproduced at the outlet of the die.

(2) When an operating terminal device of the die lip adjusting mechanismcorresponding to a portion of the film is controlled, there occurs aninterference phenomenon that thickness of an adjacent portion of thefilm to the operating terminal device of the adjusting mechanism ischanged.

(3) In order to minimize the interference effect to the film thicknessdue to mutual interference of the operating terminal device of the lipadjusting mechanism described in (2), there is a control system whichupdates commands of the operation amount for a multiplicity of operatingterminal devices simultaneously. The control system performs acalculation each time a thickness gauge which is reciprocated along thewidth of the film reaches an end of the film in which the thicknessgauge completes reading of thickness data of the film along the widththereof. Consequently, an operation until the thickness gauge reachesthe end of the film after the thickness gauge has measured thickness ofa portion of the film takes a time, which is a dead time L₂ until thecontrol system starts the calculation actually after the thickness datahas been obtained. Accordingly, a dead time after the operation amountfor the operating terminal device has been changed and its influence hasbeen detected as a thickness data until the detected thickness data isemployed to perform the calculation is a sum of the dead time L₁described in (1) and the dead time L₂ described above.

As described above, the conventional film thickness controller has (A) afirst drawback of producing a large dead time and (B) a second drawbackof generating the interference effect. Description is now made toproblems due to these drawbacks.

A. Problem due to large dead time:

FIG. 5(a) is a block diagram of a thickness control system in the casewhere the controller of FIG. 4 involves the dead times L₁ and L₂. FIG.5(b) is a block diagram of a thickness control system in which the deadtimes are combined to one equivalent time. A general feedback controlsystem does not contain such a dead time, while the thickness controlsystem contains such a large dead time (L₁ and L₂) as shown in FIG.5(b).

Consequently, since there is a large phase delay due to the dead time, again of a controller can not be increased even if phase compensation iseffected in order to attain stability in the control system.Accordingly, the high-speed response and the steady-state accuracy ofthe control system are deteriorated. Further, the thickness of the filmis always influenced by an external disturbance due to variation of anadjacent die lip adjusting mechanism.

B. Problem due to interference effect:

In FIG. 30, when an operating terminal device of a portion of theconventional adjusting mechanism 12b is operated, the thickness of aportion of the film corresponding to an adjacent operating terminaldevice is changed. Accordingly, the operating terminal device of theportion of the adjusting mechanism and the control loop for controllingthickness of a portion of film corresponding to the position of theoperating terminal device interfere with each other. Consequently, thefollowing problems occur:

(1) Even if the stability of the control loop shown in FIG. 30 isensured, since operation of an operating terminal device of theadjusting mechanism 12b is influenced by the control loop which controlsthickness of the film corresponding to an adjacent operating terminaldevice, the control loops interfere with each other and the stability ofthe whole control system is not ensured when the thickness of the filmis controlled over the whole width of the film. Accordingly, in order toeliminate the influence of the mutual interference, the gain of thecontroller 13b is reduced so that the control system has a low-speedresponse.

(2) Conversely, when it is considered to design a stable control systemconstituting a multi-variable system in consideration of the mutualinterference between the operating terminal devices of the adjustingmechanism 12b, the control system becomes a very large system since ahundred or more operating terminal devices are usually disposed in thelongitudinal direction of the slot of the die lips 4b and there aredetected values of the film thickness equal to the number of theoperating terminal devices. Accordingly, it is difficult to design sucha large system with ensured stability.

OBJECT AND SUMMARY OF THE INVENTION

It is a first object of the present invention to provide a filmthickness controller having a control device which solves the problemsdue to the dead time in a film thickness controller having a large deadtime.

It is a second object of the present invention to provide a filmthickness controller which solves the problems due to the interferenceeffect in a film thickness controller having a die lip adjustingmechanism with the interference effect by combining a plurality of basiccontrol systems.

A. SUMMARY OF FIRST INVENTION

A film thickness controller for use in an extrusion molding apparatusand a flowing type molding apparatus of film including a die having amechanism which controls a discharge amount of molten plastic along thewidth of the film and a thickness gauge for detecting a variation ofthickness of the film after the elapse of a dead time L₁ correspondingto a time required for movement of the film between the die and thethickness gauge, comprises a subtracter for producing a differencebetween a thickness value detected by the thickness gauge in apredetermined position along the width of the film and a set value ofthickness in the predetermined position, an integrator fortime-integrating the difference of thickness produced by saidsubtracter, a memory for storing past time sequence data of operationamounts of an operating terminal device during a time equal to a sum ofthe dead time L₁ and a time L₂ until the thickness gauge reaches an endof the film after detection of thickness in the predetermined position,an operational calculator for producing the past time sequence data ofoperation amounts of the operating terminal device stored in said memoryand an estimated value of a state variable at a time earlier than a timewhen the set value of the detected thickness value of film has beeninputted by a dead time L, a state shifter for receiving an output ofsaid integrator and an output of said operational calculator andmultiplying a coefficient for shifting the state by the dead time L toproduce a state estimated value at a predetermined time, a stateprediction device for receiving the past time sequence data of operationamounts of the operating terminal device stored in said memory toproduce variation of a state based on establishment of input from acertain time to a time after the lapse of the dead time L, an adder foradding an output of said state shifter and an output of said stateprediction device to produce the state estimated value at thepredetermined time, and an operation amount commander for multiplying astate estimated value at a certain time produced from said adder by astate feedback gain to produce an operation amount command value of saidoperating terminal device.

According to the first invention, a multiplicity of heaters are disposedalong the width of the film to control a temperature of molten plasticwhich is material of the film, and the thickness gauge detects actualthickness of the film at a position downstream of the flowing film andcorresponding to the position of the heater in the width direction ofthe film. A difference between the detected actual thickness and a setthickness is calculated by the subtracter and is time-integrated by theintegrator while a correction command is fed back. Thus, a temperatureof the heater is controlled and a temperature of the molten plastic iscontrolled to adjust the fluidity thereof so that thickness of the filmis always maintained within the set value. The phase delay due to thedead time is corrected by estimation of the past state corresponding tothe dead time by the operational calculator and time-integration duringthe time corresponding the past state by the state shifter and the stateprediction device.

B. SUMMARY OF SECOND INVENTION

A film thickness controller for use in an extrusion molding apparatusand a flowing type molding apparatus including a die having a slot alongwhich a plurality of operating terminal devices of a discharge amountadjusting mechanism of molten plastic are disposed and a thickness gaugefor detecting variation of thickness after the lapse of a dead timecorresponding to a time required for movement of the film between thedie and the thickness gauge, comprises a thickness data memory forstoring thickness data produced by the thickness gauge, a distributorfor receiving an output of said thickness data memory and an arrival endidentification signal which is produced by the thickness gauge toidentify whether the thickness gauge reaches either of both ends of thefilm, a plurality of basic control means for receiving an output of saiddistributor and the arrival end identification signal produced by thethickness gauge, a plurality of command value memories each receiving anoutput of each of said plurality of basic control means, a superpositionadder for receiving an output of each of said command value memories,and an operation value memory for receiving an output of saidsuperposition adder and for supplying an output of said operation valuememory to said basic control means.

According to the second invention, the following operation is attained.

(1) The thickness gauge measures thickness of the film while moving inthe reciprocating manner along the width of the film. Since the film ismoved at a certain speed, the thickness gauge measures the filmthickness along a locus as shown in FIG. 27. Accordingly, the thicknessgauge produces thickness data of a portion of the film corresponding toeach operating terminal device sequentially and also produces an arrivalend identification signal indicating whether the thickness gauge reachesone end (A) or the other end (B) when the thickness gauge reaches an endof the film.

(2) The thickness data memory stores thickness data of the film whichare measured by the thickness gauge over the whole width of the film andwhich are thickness data of each portion of the film corresponding toeach of the operating terminal devices.

(3) The distributor receives the arrival end identification signal ofthe thickness gauge and further receives the thickness data over thewhole width of the film from the thickness data memory at the same timeas receiving of the arrival end identification signal. The distributorsupplies a set of predetermined number of thickness data from thereceived thickness data to a predetermined basic control system to bedescribed later.

(4) Each of basic control systems (control means) receives the set ofthickness data supplied from the distributor and the arrival endidentification signal from the thickness gauge and further receives dataset from the operation amount memory described later to calculateoperation amount command values for a plurality of adjacent operatingterminal devices containing a predetermined operating terminal device sothat the thickness of a portion of the film corresponding to thepredetermined operating terminal device is controlled to a predeterminedvalue stably.

(5) The command value memories store the operation amount command valuesof the plurality of operating terminal devices calculated by thecorresponding basic control systems, respectively.

(6) The superposition adder receives contents of the command valuememories storing the operation amount command values of the basiccontrol systems corresponding to each of operating terminal devices andeffects superposition, addition and average operation to the commandvalues of each of the operating terminal devices to define final commandvalues of each of the operating terminal devices.

(7) The operation amount memory stores the operation amount commandvalues of each of the operating terminal devices defined by thesuperposition adder retroactively to the past by a time corresponding toa sum L(=L₁ +L₂) of the dead time L₁ of the thickness gauge and a timeL₂ required for movement of the thickness gauge from the positioncorresponding to each of the operating terminal devices to an end of thefilm.

As described above, the basic control systems can control thickness ofthe film corresponding to each of the heaters (operating terminaldevices) containing in the own systems to a predetermined value and cancontrol thickness over the whole width of the film by combination of thebasic control systems.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a controller according to a firstembodiment of the first invention;

FIG. 2 schematically illustrates a configuration of a conventional filmmanufacturing plant;

FIG. 3 is a front view showing an conventional arrangement of heatersembedded in a die;

FIG. 4 is a block diagram of a conventional film thickness controller;

FIGS. 5(a) and 5(b) are a block diagram of a conventional film thicknesscontroller containing dead time, in which FIG. 5(a) is a block diagramhaving separate blocks expressing dead times L₁ and L₂, respectively,and FIG. 5(b) is a block diagram having a combined block expressing asum of the dead times L₁ and L₂ ;

FIG. 6 illustrates a correspondence of positions of five heaters andfive thickness detection positions;

FIG. 7 is a block diagram expressing a dynamic mathematical model offilm thickness;

FIG. 8 shows a locus of a thickness gauge for detecting thickness offilm;

FIG. 9 is a diagram illustrating a time interval of calculation andtime-integration section;

FIGS. 10, 11 and 12 are diagrams illustrating various time-integrationsections;

FIGS. 13(a) to 14(b) are graphs showing simulation results using anapparatus of the first embodiment of the first invention (when a setvalue of thickness is changed and when external heat is added to aheater, respectively);

FIGS. 15 to 21 are diagrams concerning a second embodiment of the firstinvention, in which;

FIGS. 15, 16 and 17 are diagram illustrating time intervals ofcalculation and time-integration sections;

FIGS. 18(a) to 19(b) are graphs showing simulation results illustratingeffects in the case where an average dead time L is used as anintegration section of a state shifter and a state prediction device;and

FIGS. 20(a) to 21(b) are graphs showing simulation results (when a setvalue of thickness is changed and when external heat is added to aheater, respectively);

FIG. 22 is a block diagram showing a configuration of a controller of afirst embodiment of the second invention;

FIG. 23 is a block diagram expressing a dynamic mathematical model of afilm thickness manufacturing process of the first embodiment of thesecond invention;

FIG. 24 is a block diagram showing a configuration of a basic controlsystem of the embodiment;

FIG. 25 illustrates an application procedure of the basic control systemof FIG. 24 to thickness control points;

FIG. 26 illustrates a correspondence of positions of five arbitraryoperating terminal devices and five thickness detection positions of theembodiment;

FIG. 27 illustrates a locus of a thickness gauge which is reciprocatedto detect thickness of film in the embodiment;

FIG. 28 schematically illustrates a configuration of a conventional filmmanufacturing plant;

FIG. 29 illustrates an arrangement of operating terminal devicesembedded in a die of FIG. 28;

FIG. 30 is a block diagram showing a configuration of the conventionalfilm thickness controller;

FIGS. 31(a) to 34(b) are graphs showing simulation results of theembodiment when a set value of thickness is changed and FIGS. 31(b) to33(b) are graphs showing simulation results of the embodiment whenexternal heat is added to a heater;

FIG. 35 illustrates a discrete time for determining a gain matrix of anoperational calculator of the second embodiment; and

FIGS. 36(a) to 39(b) are graphs showing simulation results of the secondembodiment when a set value of thickness is changed, and FIGS. 36(b) to39(b) are graphs showing simulation results of the second embodimentwhen external heat is added to a heater.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

A1. First Embodiment of First Invention

(a) Transfer Function Matrix G(s)

In order to explain the embodiment, referring to FIG. 6, control ofthickness 3' of film to a predetermined value is considered by employingfive heaters 1 to 5 and thickness values 1' to 5' of film measured bythickness gauges 10 located corresponding to the heaters 1 to 5. Thereason that heaters 1, 2 and 3, 4 adjacent to the heater 3 areconsidered in order to control the thickness 3' is to set a controlsystem taking interference of the heaters 1, 2 and 4, 5 to the thickness3' in consideration. Although there are many heaters on both sides ofthe heaters 1 and 5, it is considered that influence to the thickness 3'by the heaters disposed outside of the heaters 1 and 5 is as small asnegligible. Amounts of heat generated by the heaters 1 to 5 are u₁ (t),u₂ (t), u₃ (t), u₄ (t) and u₅ (t), respectively, and measured values ofthickness 1' to 5' are y₁ (t), y₂ (t), y₃ (t), y₄ (t) and y₅ (t),respectively.

When Laplace transforms of u_(i) (t) and y_(i) (t) (i=1-5) are U_(i) (s)and Y_(i) (t) (i=1-5), respectively, U_(i) (s) and Y_(i) (s) are relatedto each other by the following transfer function matrix G(s): ##STR1##g₁ (s) is a transfer function which introduces temporal variation ofthickness 3' when only an amount of heat generated by the heater 3, forexample, is changed. g₂ (s) is a transfer function which introducestemporal variation of thickness 3' when only an amount of heat generatedby the heater 2 or 4 is changed. g₃ (s) is a transfer function whichintroduces temporal variation of thickness 3' when only an amount ofheat generated by the heater 1 or 5 is changed. The equation (1) doesnot contain a dead time due to movement of the film from an outlet ofthe die to the thickness gauge. Accordingly, g₁ (s), g₂ (s) and g₃ (s)are expressed by a rational function of Laplacian operator s. Further,non-diagonal items of the transfer function matrix G(s) of the equation(1) express interference to thickness by the adjacent heaters.

(b) State Equation

When the relation between the input U_(i) (s) and the output Y_(i) (s)(i=1-5) of the equation (1) is expressed, the following state equationin the canonical form which is convenient for design of the controlsystem is employed:

    x(t)=Ax(t)+Bu(t)                                           (2)

    y(t)=Cx(t)                                                 (3)

where x is a state vector, u is an input vector in which u(t)=[u₁ (t),u₂(t),u₃ (t),u₄ (t),u₅ (t)]^(T) (where T expresses transposition), and yis an output vector in which y(t)=[y₁ (t),y₂ (t),y₃ (t),y₄ (t),y₅(t)]^(T). The state equations (2) and (3) are controllable andobservable.

The state vector means a vector consisting of a set of variables inwhich a state of the system is defined when the vector is obtained.

The input vector means a set of variables which is a boundary conditionof the state equation for the state vector, and the system is controlledby controlling the input vector.

The output vector means a vector consisting of a set of measured amountsdefined by the state vector, and the system is controlled by measuringthe output vector.

The term "controllable" means that the state vector can be controlled bythe input vector.

The term "observable" means that the state vector can be found by theoutput vector.

(c) Output Equation Taking Dead Time into Consideration

Assuming that a dead time due to movement of the film from an outlet ofthe die to the thickness gauge is L₁ and a time required for movement ofthe thickness gauge from the thickness measurement point 3' to an end ofthe film is L₂, the whole dead time L of the output vector y is given,as shown in FIG. 5(b), by:

    L=L.sub.1 +L.sub.2                                         (4)

Accordingly, the output equation (3) is expressed by:

    y(t)=Cx(t-L)                                               (5)

The relation between the input u(t) (amount of heat generated by theheater) and the output y(t) (detected value of the thickness gauge) isshown in FIG. 7 on the basis of the equations (2) and (5).

(d) Arrival End Identification signal

The thickness gauge measures thickness of film while moving inreciprocating manner along the width of the film. Since the film ismoved at a certain speed, the thickness gauge measures the filmthickness along a locus as shown in FIG. 8. When the position of thethickness 3' is shown by A in FIG. 8, the dead time L₂ due to movementof the thickness gauge is expressed by a time L₂ ' corresponding tomovement between A and B in FIG. 8.

On the other hand, when the control calculation is made at an end of thefilm shown by C of FIG. 8, the dead time L₂ due to movement of thethickness gauge is expressed by a time L₂ " corresponding to movementbetween A' and C in FIG. 8. As seen from FIG. 8, since the times L₂ 'and L₂ " are different, the control system which controls the thickness3' to a predetermined value is characterized in that the dead time L ofFIG. 7 in the case where the control calculation is made at the end ofthe film shown by (B) of FIG. 8 is different from that in the case wherethe control calculation is made at the end of the film shown by (C) ofFIG. 8. Accordingly, the thickness gauge produces the arrival endidentification signal d indicating an end at which the thickness gaugearrives.

(e) Integrator and Output X_(I) (t) thereof

In order to avoid influence of external disturbance due to thermalconduction from an adjacent heaters to control the thickness 3' to a setvalue, an integrator is introduced to integrate deviation ε(t)=r₃ (t)-y₃(t) between the detected value y₃ (t) of the thickness 3' and the setvalue r₃ (t). In the following description, the set value r₃ (t)=0.

the integrator integrates the deviation ε(t) until the current time t.However, the deviation can be actually integrated only until the time(t-L) because of the dead time L. Accordingly, an output X_(I) (t) ofthe integrator is expressed by the following equation: ##EQU1## where C₃expresses the third line of C matrix of the equation (3).

(f) Augmented System State Vector X(t)

The first term of the right side of the equation (6) is time-integrationof a value capable of being obtained from the thickness gauge until timet and accordingly it can be calculated. However, the integrated value ofthe second term of the right side can not be obtained and the timeintegration can not be calculated as it is. Accordingly, in order toobtain prediction of X_(I) (t) at time t, an augmented system as followsin which X_(I) (t) is contained in the state variable is considered.From the equation (6), the following equation is obtained:

    X.sub.I (t)=-C.sub.3 X(t-L)-C.sub.3 X(t)+C.sub.3 X(t-L)=-C.sub.3 X(t)(7)

From the equations (2) and (7), ##EQU2## By using the state vectorX(t)=[X_(I) (t), X(t)]^(T) of the augmented system, the equation (8) isexpressed as follows: ##EQU3##

(g) Feedback Gain Matrix

If the state feedback gain matrix for the equation (9) is F=[f₁, F₂ ],the input u(t) is given by ##EQU4## where f₁ expresses the first columnof F matrix. If the feedback gain matrix F is defined so that allcharacteristic values of matrix (A-BF) are in the stable region if X_(I)(t) and X(t) are obtained, the thickness y₃ (t) can be controlled to apredetermined value on the basis of the input u(t) stably. Further,since the matrices A and e,ovs/B/ are not influenced by the dead time,this design method can determine the feedback gain matrix F as if it isa system having no dead time L and can obtain the high-speed responseand the steady-state accuracy of the control system.

(h) Calculation of X_(I) (t) and X(t)

The problem is whether X_(I) (t) and X(t) can be calculated or not. IfX_(I) (t) and X(t) at the current time t can not be obtained, the stablecontrol can not be obtained in the case of the above mentioned feedbackgain matrix F, and the high-speed response and the steady-state accuracyof the control system are both deteriorated. The problem (2) in theprior art can not be solved.

X_(I) (t) and X(t) are obtained as shown in the equation (12) byinitializing the time (t-L) and integrating the equation (9) from thetime (t-L) to the time t. Since the input u(t) is already known, thestate values X_(I) (t) and X(t) are estimated by performing theintegration retroactively to the past by the time L. ##EQU5##

(i) Calculation of X_(I) (t-L) and X(t-L)

X_(I) (t-L) of the first term of the right side of the equation (12) isexpressed on the basis of the equation (7) as follows: ##EQU6## Sincethe right side of the equation is calculable and is an integrated valueof control deviation of the output y₃ (t) at the current time t, theequation (13) is expressed by:

    X.sub.I (t-L)=X.sub.I (t)                                  (14)

where X_(I) (t) is an integrated value of control deviation of thedetected value y₃ (t) of the thickness 3'.

X(t-L) can be estimated as follows: From the equations (2) and (5),

    X(t-L)=Ax(t-L)+Bu(t-L)                                     (15)

    y(t)=Cx(t-L)                                               (16)

A variable ω(t) defined by the following equation is introduced.

    ω(t)=X(t-L)                                          (17)

From the equations (15) to (17), the following equations are obtained.

    ω(t)=Aω(t)+Bu(t-L)                             (18)

    y(t)=Cω(t)                                           (19)

The operational calculator for the equations (18) and (19) is designedto obtain an estimated value X(t-L)=ω(t) from the detected thicknesssignal y(t).

(j) Dead Time L and Calculation Period

Since the calculation is performed each time the thickness gauge reachesthe point B or C as shown in FIG. 8, that is, at regular intervals oftime T.

The relation of the dead time L and the period T of performing thecontrol calculation is described. It is assumed that the position A ofthe thickness 37 exists near the end C of the film as shown in FIG. 8.When the control calculation is made at the end B of the film, the wholedead time L of the equation (4) is large since the dead time L₂ ' islarge. On the other hand, when the control calculation is made at theend C of the film, the whole dead time L is small since the dead time L₂" is small.

In the embodiment, the dead time L is classified into the following twocases. A case to which the dead time L belongs is determined by thearrival end identification signal produced when the thickness gaugereaches the end of the film.

Case 1: 2T≦L<3T

Case 2: T<L<2T

(k) Discrete Equation

It is necessary to change the equations (18) and (19) to discreteequations for each time interval T and design the operationalcalculator.

(1) Case 1 (2T≦L<3T)

As shown in FIG. 9, it is assumed that the control calculation isperformed at time t_(k-3) to t_(k-1). It is assumed that at the timet_(k+1), the output vector y(k+1) is obtained as a set of thickness dataand the input vector u(k) is maintained constant during time t_(k) tot_(k+1). From the equation (18), the following equation is derived.##EQU7## If the following variable is introduced, the equation (20) isexpressed by the equation (21). ##EQU8## The integration of the rightside of the equation (21) means that the double-line portion of FIG. 10is integrated. Accordingly, the equation (21) is expressed by ##EQU9##The following variables φ, Γ₁, Γ₂ are introduced. ##EQU10## If thediscrete value ω(t_(k)) is expressed by ω(k), the equation (22) isexpressed by

    ω(k+1)=φω(k)+Γ.sub.1 u(k-3)+Γ.sub.2 u(k-2)(27)

The discrete equation of the equation (19) is given by

    y(k+1)=Cω(k+1)                                       (28)

By designing the operational calculator for the equations (27) and (28),the estimated value ω(k+1) at time t=t_(k+1) is obtained from thefollowing two equations.

    ω(k+1)=φω(k)+Γ.sub.1 u(k-3)+Γ.sub.2 u(k-2)(29)

    ω(k+1)=ω(k+1)+K[y(k+1)-Cω(k+1)]          (30)

where K is a feedback gain matrix of the operational calculator.

According to the equations (29) and (30), the state ω(k+19 at timet=t_(k+1) can be estimated from the set of thickness data y(k+1) at timet=t_(k+1). The estimated error ω(k)=ω(k)-ω(k) at this time is expressedby

    ω(k+1)=(φ-KCφ)ω(k)                     (31)

Accordingly, if the gain matrix K of the operational calculator isdefined so that all the eigen values of matrix (φ-KCφ) exist in thestable region, the estimated error can be reduced with the lapse oftime.

(2) Case 2 (T<L<2T)

As shown in FIG. 10, it is assumed that control calculation is performedat time t_(k-2), t_(k-1), t_(k), and t_(k+1). The integration of theright side of the equation (21) means that the double-line portion ofFIG. 10 is integrated. The discrete equation at this time is expressedby

    ω(k+i)=φω(k)+Γ.sub.1 u(k-2)+Γ.sub.2 u(k-1)(31)

m of the equations (24) and (25) is given by

    m=2T-L                                                     (32)

The estimated value ω(k+1) at time t=t_(k+1) is obtained from thefollowing two equations.

    ω(k+1)=φω(k)+Γ.sub.1 u(k-2)+Γ.sub.2 u(k-1)(33)

    ω(k+1)=ω(k+1)+K[y(k+1)-Cω(k+1)]          (34)

The equation of the estimated error is the same as the equation (31) andthe same thing as the case 1 is applicable in order to reduce theestimated error with the lapse of time.

From the foregoing, the estimated value of the state X(t_(k+1) -L) att=t_(k+1) can be obtained in the following sequence.

(1) If time t=t_(k+1) is a termination time of the calculation executionperiod T and it is understood from the arrival end identification signalproduced from the thickness gauge that the thickness gauge reaches theend (B) of the film as shown in FIG. 8, ω(k+1) is calculated from theequations (29) and (30) and the estimated value x(t_(k+1) -L)=ω(k+1) ofx(t_(k+1) -L) is obtained.

(2) If time t=t_(k+1) is a termination time of the control calculationexecution period T and it is understood from the arrival endidentification signal produced from the thickness gauge that thethickness gauge reaches the end (C) of the film as shown in FIG. 8,ω(k+1) is calculated from the equations (33) and (34) and the estimatedvalue x(t_(k+1) -L)=ω(k+1) is obtained.

(1) Calculation of Second Term of Equation (12)

The final thing to do is to obtain the integration term of the rightside of the equation (12), that is, ##EQU11## The integration I is topredict variation of the state ##EQU12## by the input u(t) from time(t-L) to time t. At this time, the dead time L is classified to thefollowing two cases. A case to which the dead time L belongs isdetermined by the arrival end identification signal produced from thethickness gauge.

Case 1: 2T≦L<3T

Case 2: T<L<2T

(1) Case 1: (2T≦L<3T)

In the integration I, the double-line portion of FIG. 11 is integrated.##EQU13## If the following variable is introduced, the integration I isexpressed by the equation (37). ##EQU14##

(2) Case 2 (T<L<2T)

In the integration I, the double-line portion of FIG. 12 is integrated.##EQU15## If the variable η of the equation (36) is introduced, theintegration I is expressed by ##EQU16##

(m) Estimated Value of State Value [X_(I) (t), X(t)]^(T)

From the equations (12), (14), (29), (30), (33), (34), (37) and (38),the estimated value [X_(I) (k+1), X(k+1)]^(T) of the state value [X_(I)(t), X(t)]^(T) at current time t=t_(k+1) is obtained from the followingequation. ##EQU17##

(n) Means for Executing Calculation

FIG. 1 is a block diagram of a controller implementing the firstinvention. In the first embodiment, each of blocks is operated asfollows.

(1) The detected value y(k+1) of film thickness (vector consisting of y₁(k+1, y₂ (k+1), y₃ (k+1), y₄ (k+1) and y₅ (k+1)) is obtained through athickness gauge 10 and a sampler 100 at the control calculationexecution time t=t_(k+1) of the time interval T. The sampler 100 closesfor each calculation execution time t=t_(k+1), that is, the sampler 100closes each time the thickness gauge 10 reaches the end B or C of thefilm shown in FIG. 8. Further, when the thickness gauge 10 reaches theend B or C of the film, the gauge 10 produces the arrival endidentification signal d which indicates the end to which the gauge hasreached.

(2) The detected value y₃ (k+1) of the detected film thickness valuey(k+1) is supplied to a subtracter 101 which produces thicknessdeviation ε(k+1)=r₃ (k+1)-y₃ (k+1) between the detected value y₃ (k+1)and a set value of thickness r₃ (k+1).

(3) The integrator 102 is supplied with the thickness deviation ε(k+1)from the subtracter 101 and produces a time-integrated value of thethickness deviation from the following equation.

    X.sub.I (k+1)=X.sub.I (k)+0.5(t.sub.k+1 -t.sub.k){ε(k)+ε(k+1)}                    (40)

where ε(k) is a thickness deviation at the last thickness detection time(t=t_(k)) and X_(I) (k) is an output of the integrator 102 at t=t_(k).

The integrator 102 includes a function of an external disturbancecompensator and serves to compensate external heat varying the thicknessy₃ with heat generated by the heater so that the thickness y₃ is alwaysmaintained to be a set value.

(4) When the thickness gauge reaches either end of the film, thethickness gauge produces the arrival end identification signal d. ω(k+1)is calculated from the equations (29) and (30) or (33) and (34) inresponse to the identification signal d. More particularly, in theequation (29) and (30) for the past time sequence data of heat generatedby the heater stored a memory 104, u(k-3) and u(k-2) are supplied to theoperational calculator, while in the equations (33) and (34), u(k-2) andu(k-1) together with the detected film thickness value y(k+1) aresupplied to the operational calculator, which produces an estimatedvalue X(t_(k+1) -L)=ω(k+1) of the state variable at time t(_(k+1) -L)earlier than time t_(k+1) by the dead time L determined by the arrivalend identification signal d produced by the thickness gauge.

(5) In the calculation of the first term of the right side of theequation (39), the state estimated value [X_(I) (k+1), ω(k+1)]^(T) attime (t_(k+1) -L) is multiplied by a coefficient e^(AL) for shifting thestate by the dead time L to obtain the state estimated value e^(AL)[X_(I) (k+1), ω(k+1)]^(T) at time t_(k+1). That is, the output X_(I)(k+1) of the integrator 102 and the output ω(k+1) of the operationalcalculator 103 are supplied to state shifter 105, which multiplies themby the coefficient for shifting the state by the dead time L determinedby the arrival end identification signal d of the thickness gauge toobtain the state estimated value at time t_(k+1). Since the magnitude ofthe dead time L is different depending on the end of the film which thethickness gauge reaches, the coefficient e^(AL) is different dependingto the position of the thickness gauge upon control calculationexecution, that is, the arrival end identification signal d of thethickness gauge.

The state shift by the input u(k) applied in time domain for only thedead time L is expressed by the second term I(k+1) of the right side ofthe equation (39) and correction therefor is made by a state predictiondevice 106.

(6) The second term I(k+1) of the right side of the equation (39)expresses an amount of shift of states for time sequence input datau(k-2), u(k-1) and u(k) applied to the time domain from time (t_(k+1)-L) to time t_(k+1). I(k+1) is calculated from the equation (37) or (38)depending on the end of the film which the thickness gauge reaches, thatis, depending on the arrival end identification signal produced by thethickness gauge. More particularly, the past time sequence data of theheat generated by the heater (in this case, three data of u(k-2), u(k-1)and u(k)) determined by the magnitude of the dead time L stored in thememory 104 are supplied to the state prediction device 106 and the statevariation amount I(k+1) by the input u(k) from time (t_(k+1) -L) to timet_(k+1).

(7) Output e^(AL) [X_(I) (k+1), ω(k+1)]^(T) of the state shifter 105 andoutput I(k+1) of the state prediction device are added in adder 107which produces the state estimated value [X_(I) (k+1), X(k+1)]^(T) attime t_(k+1). Thus, although the operational calculator 103 can obtainonly the state estimated value at time t_(k+1) -L due to the dead timeL, the state estimated value at time t_(k+1) can be obtained byintegration in the state shifter 105 and the state prediction device 106for the dead time L. Influence of phase delay due to the dead time L canbe eliminated by this operation.

(8) An amount u(k+1) of heat generated by the heater from time t_(k+1)to next time t_(k+2) for control calculation is defined by the followingequation using state feedback gain (f₁, F₂).

    u(k+1)=-f.sub.1 X.sub.I (K+1)-F.sub.2 X(k+1)               (41)

The adder 107 supplies the state estimated value [X(k+1), X(k+1)]^(T) attime t_(k+1) to a commander 108 for heat generated by the heater. Thecommander 108 multiplies the state estimated value [X(k+1),X(k+1)]^(T)by the state feedback gain to define a command value of heat generatedby the heater.

(9) The above control calculation is executed after the next detectedvalue y(k+2) of film thickness is obtained from the sampler 100 at timet=t_(k+2) of calculation execution when the thickness gauge is movedalong the width of the film after the time period T and reaches theopposite film end.

(0) Example of Design

As a first actual example, an example of design is described in the casewhere transfer functions g₁ (S), g₂ (S) and g₃ (S) are given by thefollowing equations: ##EQU18## u_(i) (t)(i=1-5) is variation (Kcal/s) ofheat generated by the heater, and y_(i) (t)(i=1-5) is variation (cm) ofthickness of film at the position of the thickness gauge correspondingto the position of the heater. The dead time L₁ due to movement of thefilm and times L₁ ' and L₂ " required for movement of the thicknessgauge from the thickness control point 3' to the film end assume thefollowing values.

    L.sub.1 =30 seconds

    L.sub.2 '=15 seconds

    L.sub.2 "=1.5 seconds

It is assumed that the thickness control point 3' exists at the end C ofthe film as shown in FIG. 8. The control calculation execution period Tassumes the following value.

    T=16.5 seconds                                             (45)

In order to design the control system, it is necessary to express therelation between the input u(t) and the output y(t) of the equation (1)and obtain the controllable and observable state equations (2) and (3).G(s) constituted of g₁ (s), g₂ (s) and g₃ (s) of the equations (42) to(44) can be expressed by an equation of the 77th degree, while thecontrollable and observable equation has been found to be an equation ofthe 39th degree. Accordingly, the equations (2) and (3) of the 39thdegree are obtained from G(s).

(1) Decision of State Feedback Gain Matrix F

The state feedback gain matrix F of the equation (11) is obtained as asolution of an optimum regulator problem for the state equation (8)extended to the equation of the 40th degree on the basis of the equation(2). Since the equation (8) is a state equation of a continuous timesystem, the equation is changed to a discrete state function with thesampling period T=16.5 seconds and a regulator solution is applied. Aproper estimation function is employed to obtain the state feedback gainmatrix F and as a result the following values are obtained as the eigenvalues of the matrix (A-BF).

    0.876±0.02i, 0.79, 0.50±0.07i, 0.60±0.09i, 0.60±0.06i, 0.51

Further, 30 eigen values other than above are not described since theabsolute value thereof is less than 0.1 and attenuation is fast. Sinceall eigen values are within a circle having a radius of 1, stablecontrol can be attained. Since the eigen value having the slowestattenuation is 0.88±0.02i, the stabilization time Ts can be predicted asabout 10 minutes from (0876)³⁵ ≈0.01 with definition of control error 1%as follows.

    Ts=T×35=16.5×35 sec.=577.5 sec.=9.6 min.

(2) Decision of Feedback Gain K of Operational Calculator

The feedback gain matrix K of the operational calculator of the equation(30) or (34) is obtained for the state equation (27) or (31) of the 39thdegree and the output equation (28) of the fifth degree. The gain matrixK is obtained as a solution of the optimum regulator problem so that thematrix {φ^(T) -(Cφ)^(T) K^(T) } has a stable eigen value. As a result ofobtaining the gain matrix K using a proper estimation function, thefollowing values are obtained as eigen values of the matrix (φ-KCφ).##EQU19##

30 eigen values other than above concentrate to the origin. Since allthe values are within a circle having a radius of 1, the estimated errorcan be reduced with the lapse of time. Since the eigen value having theslowest attenuation is 0.9077, the time To required for attenuation ofthe estimated error to an initial 1% can be predicted from (0.9077)⁴⁵≈0.01 as follows.

    To=T×45=16.5×45 sec.=742.5 sec.=12.4 min.

(p) Simulation Example (1)

FIG. 13 shows an example of simulation result obtained by calculationusing the gain matrices F and K obtained above. FIG. 13(a) showsvariations (variations of detected values of the thickness gauge) offive thickness values y₁ to y₅ versus time when the set value ofthickness y₃ is changed stepwise by 0.02 mm. FIG. 13(b) shows variationsof amounts U₁ to U₅ of heat generated by the heaters in the samecondition as FIG. 13(a).

Since calculation is made after the execution period of 16.5 seconds ofcalculation after the set value of thickness has been changed, variationof heat generated by the heater occurs after 16.5 seconds from change ofthe set value of thickness. An amount of heat generated by the heater ismaintained to the same value until 16.5 seconds elapse and the nextcalculation is made. The calculation is made on the basis of a newlydetected value of thickness after 16.5 seconds to change an amount ofheat generated by the heater. Accordingly, an amount of heat generatedby the heater changes stepwise as shown in FIG. 13(b).

On the other hand, variation of the detected thickness value is detectedafter the lapse of the dead time L of 31.5 seconds after the amount ofheat generated by the heater has been changed after the lapse of 16.5seconds from the change of the set value. That is, variation ofthickness is detected after the lapse of 16.5+31.5=48 seconds after theset value of thickness has been changed. Thickness y₃ is exactly changedto a set value and the change is substantially symmetrical to thethickness y₃. Variation of heat generated by the heater U₃ is largest,variations by the heaters U₂ and U₄ are largest next to the heater U₃,and variations of the heaters U₁ and U₅ are smallest.

The stabilization time is about 18.5 minutes which is considerably largeas compared with the stabilization time of 12.4 minutes calculated bythe eigen value of the operational calculator (the stabilization time bythe eigen value of the regulator is still shorter). This is based on thefollowing reason.

In order to prevent the command value of heat generated by the heaterfrom being changed largely for each calculation, the command value isdefined with weight added as follows.

    U.sub.d.k =WU.sub.d.k-1 +(1-W)U.sub.k                      (46)

where

U_(d).k : command value of heat defined by the calculation time t=t_(k),

U_(d).k-1 : command value of heat defined by the last calculation timet=t_(k-1),

U_(k) : command value of heat calculated at the calculation timet=t_(k), and

W: weight coefficient.

In this simulation, W=0.8. This means that when the calculation periodT=16.5 seconds is considered, a time delay corresponding to a delay offirst order having a time constant of 74.65 seconds is added to the heatcommander. Accordingly, it is considered that the stabilization time ofthickness control of FIG. 13 is larger than the stabilization timeestimated by the eigen value of the operational calculator. Then, evenif the thickness control is in the stabilization state, the commandvalue of heat changes for each calculation. The reason is because themagnitude of the dead time L of the first term e^(AL) of the right sideof the equation (39) which is one of the calculation equations isdifferent in one end B and the other end C of the film as shown in FIG.8.

When the present control system is applied actually, the samecalculation equation as that applied to the thickness y₃ is applied toeach of thicknesses y₁, y₂, y₄, and y₅ and each command value of heatmay be produced as a sum of results of the calculation equations. Itwill be understood that the control system considerably reducesinfluence of the dead time since the time required for stabilizingvariation of thickness when heat generated by the heater is changedstepwise without control of heat is about 10 minutes.

(q) Simulation Example (2)

The second actual example is now described with reference to FIG. 14,which shows a control result when external heat of 8.4 wattage isapplied to the heater u₃. FIG. 14(a) shows variations of thicknessvalues y₁ to y₅ versus time, and FIG. 14(b) shows variations of heat u₁to u₅ generated by the heaters versus time. As shown in FIG. 14(a),although the thickness y₃ is once increased by the external heat of theheater u₃, the thickness y₃ is returned to the original set value bychanging the amounts of heat generated by the heaters u₁ to u₅ and thestabilization time is about 18.5 minutes in the same manner as FIG. 13.It is understood that variation due to the external disturbance isexactly compensated by introducing the integrator in the present controlsystem.

The thickness values y₂ and y₄ are once increased by influence ofexternal heat through thermal conduction along the width of the die. Thethickness values y₁ and y₅ are also influenced similarly, although theinfluence is small as compared with y₂ and y₄. In order to cancel theinfluence of such external heat, reduction of heat generated by theheater u₃ is largest, reduction by the heaters u₂ and u₄ is largest nextto the heater u₃, and reduction by the heaters u₁ and u₅ is smallest.When external heat is applied to the heater u₃, other thickness valuesy₁, y₂, y₄ and y₅ are also changed, although such interference effectcan be canceled by applying the same calculation equation as that forthe thickness value y₃ to each of the thickness values y₁, y₂, y₄ andy₅.

A2. Second Embodiment of First Invention

(a) Relation to First Embodiment of First Invention

In the second embodiment, the process that the same elements as in thefirst embodiment are utilized, the equations (1) to (19) are derived,the operational calculator for the equations (18) and (19) is designedand the estimated value X(t-L)=ω((t) of X(t-L) is obtained from thedetected value of thickness y(t) is quite identical with that of thefirst embodiment.

In the second embodiment, a known ω(t_(k)) obtained by the calculationperformed in the step just before the current step is used to obtainω(t_(k+1)).

(b) Dead Time

In the second embodiment, the calculation is also executed each time thethickness gauge reaches the end B or C of the film as shown in FIG. 8.That is, the calculation is executed at regular intervals of time T. Thetime T is a time required for movement of the thickness gauge along thewidth of the film.

On the other hand, the dead time L for the position A of thickness 3 isdifferent depending on the end B and the end C. That is, ##EQU20## It isapparent that L_(B) >L_(C) for the position A. In the description below,it is assumed that L_(B) and L_(C) satisfy T<L<2T.

(c) Known ω(t_(k))

FIG. 15 is a diagram for explaining the calculation for the end B forobtaining the estimated value ω(t_(k+1) -L_(B)). It is assumed that thecalculation is made at time t_(k-2) to t_(k+1). Further, it is assumedthat the input u(k) of the heater is maintained to constant from t_(k)to t_(k+1).

It is assumed that the thickness gauge reaches the end B at timet_(k+1). Accordingly, it is considered that the thickness gauge hasreached the end C at the past time t_(k) before time t_(k+1) by time T.Thus, it is assumed that the estimated value ω(t_(k))=X(t_(k) -L_(C))has been obtained in the calculation for the end C executed at timet_(k). In the calculation for the end B executed at time t_(k+1), theknown ω(t_(k)) is employed to obtain the estimated valueω(t_(k+1))=X(t_(k+1) -L_(B)).

FIG. 16 is a diagram for explaining the calculation for the end C forobtaining the estimated value X(t_(k+1) -L_(C)). It is assumed that thethickness gauge reaches the end C at time t_(k+1) and the thicknessgauge has reached the end B at the past time t_(k) earlier than timet_(k+1) by time T. In the calculation for the end C executed at timet_(k+1), the known ω(t_(k)) is employed to obtain the estimated valueω(t_(k+1))=X(t_(k+1) -L_(C)).

As seen from FIGS. 15 and 16, the time interval T of the calculation isconstant, while since the dead times L_(B) and L_(C) for the ends B andC, respectively, are different, the time difference expressing theestimated values ω(t_(k)) and ω(t_(k+1)) is different from the timeinterval T. Accordingly, the estimated value ω(t_(k+1)) is obtained fromthe equations (18) and (19) as follows.

(d) Discrete Equation (18) for the End B

The calculation for the end B shown in FIG. 15 is first considered. Theknown estimated value ω(t_(k))=X(t_(k) -L_(C)) is expressed by X(to).The estimated value ω(t_(k+1))=X(t_(k+1) -L_(B)) to be obtained isexpressed by X(t₁). The estimated value X(t₁) of state variable at timet₁ is estimated from the equation (18) on the basis of the known X(to)and inputs after time to. ##EQU21## When a new variable η=t₁ -τ isintroduced, the estimated value X(t₁) is transformed as follows:##EQU22## Since the integration of the right side of the equation (49)means that the double-line portion of FIG. 15 is integrated, theequation (49) is expressed by the following equation. ##EQU23## whereu(k-1) is a heater input from time t_(k-1) to t_(k), and u(k-2) is aheater input from time t_(k-2) to t_(k-1). If X(t₁)=ω(t_(k+1)) andX(t_(o))=ω(t_(k)), the equation (50) is expressed by ##EQU24## where thediscrete value ω(t_(k)) is expressed by ω(k).

(e) Discrete Equation (18) for the End C

The calculation for the end C shown in FIG. 16 is considered. At thistime, in the equation (49)

    t.sub.o =t.sub.k -L.sub.B, t.sub.1 =t.sub.k+1 -L.sub.C, t.sub.1 -t.sub.o =T-L.sub.C +L.sub.B

By integrating the double-line portion of FIG. 16, the followingequation is obtained. ##EQU25## If X(t₁)=ω(t_(k+1)) andX(t_(o))=ω(t_(k)), the estimated value ω(t_(k+1)) is given from theequation (52) by ##EQU26##

(f) Discrete Equation for Equation (19)

The discrete equation for the equation (19) is given by

    y(k+1)=Cω(k+1)                                       (54)

(g) Calculation of Estimated Value ω(k+1)

By designing the operational calculator in accordance with the equations(51), (53) and (54), the estimated value ω(t_(k+1)) at t=t_(k+1) isobtained as follows.

The calculation equation of the estimated value ω(k+1) for the end B isgiven by ##EQU27## where φ_(B) =e^(A) (T-L_(B) +L_(C))

K_(B) =gain matrix of the operational calculator.

The calculation equation of the estimated value ω(k+1) for the end C isgiven by ##EQU28## where φ_(c) =e^(A) (T-L_(C) +L_(B))

K_(C) =gain matrix of the operational calculator.

According to the equations (55) and (58), the state variable ω(k+1) att=t_(k+1) can be estimated by a set of thickness data y(k+1) att=t_(k+1).

(h) Estimated Error ω(k)

At this time, the estimated error ω(k)=ω(k)-ω(k) is expressed by thefollowing equation:

The estimated error ω(k+1) for the end B is given by

    ω(k+1)=[φ.sub.B -K.sub.B Cφ.sub.B ]ω(k)(59)

The estimated error ω(k+1) for the end C is given by

    ω(k+1)=[φ.sub.C -K.sub.C Cφ.sub.C ]ω(k)(60)

Accordingly, if the gain matrices K_(B) and K_(C) of the operationalcalculator are defined so that all eigen values of the matrices [φ_(B)-K_(B) Cφ_(B) ] and [φ_(C) -K_(C) φ_(C) ] are in the stable domain, theestimated value can be reduced with the lapse of time.

(i) Summary of Calculation of Estimated Value of ω(k+1)

From the foregoing, the estimated value of the state X(t_(k+1) -L) att=t_(k+1) can be obtained in accordance with the following sequence.

(1) When t=t_(k+1) is a termination time of the period T of calculationexecution and it is discriminated on the basis of the arrival endidentification signal produced by the thickness gauge that the thicknessgauge has reached th end B of the film shown in FIG. 8, ω(k+1) iscalculated from the equations (55) and (56) and the estimated valueX(t_(k+1) -L_(B))=ω(k+1) of X(T_(k+1) -L_(B)) is obtained.

(2) When t=t_(k+1) is a termination time of the period T of calculationexecution and it is discriminated on the basis of the arrival endidentification signal produced by the thickness gauge that the thicknessgauge has reached th end C of the film shown in FIG. 8, ω(k+1) iscalculated from the equations (57) and (58) and the estimated valueX(t_(k+1) -L_(C))=ω(k+1) is obtained. Thus, the first term of the rightside of the equation (12) can be calculated.

(j) Integration of Second Term of Equation (12)

The final thing to do is to obtain the integration term of the rightside of the equation (12), that is, ##EQU29## This integration term I isto predict variation of the state ##EQU30## by the input u(t) from time(t-L) to time t.

The integration I is to integrate the double-line portion of FIG. 17.##EQU31## If a new variable η=t_(k+1) -τ is introduced, the integrationI(k+1) is expressed by ##EQU32## Since the dead time L is differentdepending on the calculation for the ends B and C, the equation (61) isdescribed as follows.

The integration I_(B) (k+1) for the end B is given by ##EQU33##

The integration I_(C) (k+1) for the end C is given by ##EQU34##

(k) Estimated Value [X_(I) (k+1),X(k+1)]^(T)

From the equations (12), (14), (55), (56), (57), (58), (62) and (63),the estimated value [X_(I) (k+1),X(k+1)]^(T) of the state value [X_(I)(t),X(t)]^(T) at the current time t=t_(k+1) is obtained by the followingequations.

The estimated value for the end B is given by ##EQU35##

The estimated value for the end C is given by ##EQU36##

(l) Discontinuity of Estimated Value [X_(I) (k+1),X(k+1)]^(T)

When the calculation equation for the estimated value [X_(I)(k+1),X(k+1)]^(T) of the state value at time t_(k+1) is changeddepending on the end B or C as described in the equation (64) and (65),since the dead times L_(B) and L_(C) are different and change stepwise,the estimated value is not continuous each time the equation is changed.When the dead time L_(B) is larger than the dead time L_(C), theestimated value of the equation (64) is larger than that of the equation(65) and accordingly the estimated value by the equation (11) repeatedlychanges in the pulse manner. FIG. 18 shows a simulation result when theestimated value is calculated using the equations (64) and (65). In FIG.18, the time interval of calculation t=22.5 seconds, the dead time L_(B)=39 seconds and L_(C) =37 seconds. FIG. 18 shows a control result whenexternal heat of 8 W is applied to the heater U₃ stepwise. FIG. 18(a)shows variations of thickness values y₁ to y₅ versus time, and FIG.18(b) shows variations of amounts U₁ to U₅ of heat generated by theheaters versus time. As shown in FIG. 18(a), the thickness y₃ is onceincreased by the external heat, although the thickness y₃ is returned tothe original set value by changing the amounts of heat generated by theheaters U₁ to U₅. However, heat generated by the heaters is repeatedlychanged in the steady state and the thickness is also slightly changedrepeatedly. When the position of thickness y₃ approaches the end of thefilm, since a difference between the dead times L_(B) and L_(C) isincreased, a width of variation of heat generated by the heater isincreased and variation of thickness is also larger when the estimatedequations (64) and (65) are employed.

(m) Average Value L of Dead Time

In order to improve the above drawback, an average value of L_(B) andL_(C), that is, L=(L_(B) +L_(C))/2 is employed as the dead time used inthe equations (64) and (65). The equation of the estimated can be usedin common for the ends B and C. ##EQU37##

(n) Simulation Example

FIG. 19 shows a simulation result when the equations (66), (67) and (68)are used as the equation of the estimated value with the same conditionas in FIG. 18. Variation in the steady state of heat generated by theheater is eliminated.

(o) Means for Executing Calculation

FIG. 1 is a block diagram of a controller implementing the firstinvention. In the second embodiment, each of blocks is operated asfollows.

(1) The detected value y(k+1) of film thickness (vector consisting of y₁(k+1, y₂ (k+1), y₃ (k+1), y₄ (k+1) and y₅ (k+1)) is obtained through thethickness gauge 10 and the sampler 100 at the calculation execution timet=t_(k+1) of the time interval T. The sampler 100 closes for eachcalculation execution time t=t_(k+1), that is, the sampler 100 closeseach time the thickness gauge 10 reaches the end B or C of the filmshown in FIG. 8. Further, when the thickness gauge 10 reaches the end Bor C of the film, the gauge 10 produces the arrival end identificationsignal d which indicates the end which the gauge has reached.

(2) The detected value y₃ (k+1) of the detected film thickness valuey(k+1) is supplied to a subtracter 101 which produces thicknessdeviation ε(k+1)=r₃ (k+1)-y₃ (k+1) between the detected value y₃ (k+1)and a set value of thickness r₃ (k+1).

(3) The integrator 102 is supplied with the thickness deviation ε(k+1)from the subtracter 101 and produces a time-integrated value of thethickness deviation from the following equation.

    X.sub.I (k+1)=X.sub.I (k)+0.5(t.sub.k+1 -t.sub.k){ε(k)+ε(k+1)}                    (69)

where ε(k) is thickness deviation at the last thickness detection time(t=t_(k)) and X_(I) (k) is an output of the integrator 102 at t=t_(k).

The integrator 102 includes a function of an external disturbancecompensator and serves to compensate external heat varying the thicknessy₃ with heat generated by the heater so that the thickness y₃ is alwaysmaintained to be a set value.

(4) When the thickness gauge reaches either end of the film, thethickness gauge produces the arrival end identification signal d. ω(k+1)is calculated from the equations (55) and (56) or (57) and (58) inresponse to the identification signal d. More particularly, the pasttime sequence data u(k-2) and u(k-1) of heat generated by the heaterstored a memory 104 together with the detected film thickness valuey(k+1) are supplied to the operational calculator, which produces anestimated value X(t_(k+1) -L)=ω(k+1) of the state variable at timet(_(k+1) -L) earlier than time t_(k+1) by the dead time L determined bythe arrival end identification signal d produced by the thickness gauge.

(5) In the calculation of the first term of the right side of theequation (66), the state estimated value [X_(I) (k+1), ω(k+1)]^(T) attime (t_(k+1) -L) is multiplied by a coefficient e^(AL) for shifting thestate by the average dead time L defined by the equation (68) to obtainthe state estimated value e^(AL) [X_(I) (k+1), ω(k+1)]^(T) at timet_(k+1). That is, the output X_(I) (k+1) of the integrator 102 and theoutput ω(k+1) of the operational calculator 103 are supplied to stateshifter 105, which multiplies them by the coefficient for shifting thestate by the average dead time L to obtain the state estimated value attime t_(k+1). The magnitude of the dead time L adopts the average valueof the dead times for both ends of the film as described by the equation(68).

The state shift by the input u(k) applied in time domain for only theaverage dead time L is expressed by the second term I(k+1) of the rightside of the equation (66) and correction therefor is made by a stateprediction device 106.

(6) The second term I(k+1) of the right side of the equation (66)expresses an amount of shift of states for time sequence input datau(k-1) and u(k) applied to the time domain of the average dead time fromtime (t_(k+1) -L) to time t_(k+1). I(k+1) is calculated from theequation (67) using the average dead time L. More particularly, the pasttime sequence data of the heat generated by the heater (in this case,two data of u(k-1) and u(k)) determined by the magnitude of the deadtime L stored in the memory 104 are supplied to the state predictiondevice 106 and the state variation amount I(k+1) by the input u(k) fromtime (t_(k+1) -L) to time t_(k+1).

(7) Output e^(AL) [X_(I) (k+1), ω(k+1)]^(T) of the state shifter 105 andoutput I(k+1) of the state prediction device are added in adder 107which produces the state estimated value [X_(I) (k+1), X(k+1)]^(T) attime t_(k+1). Thus, although the operational calculator 103 can obtainonly the state estimated value at time t_(k+1) -L due to the dead timeL, the state estimated value at time t_(k+1) can be obtained byintegration in the state shifter 105 and the state prediction device 106for the dead time L. Influence of phase delay due to the dead time L canbe eliminated by this operation.

(8) An amount u(k+1) of heat generated by the heater from time t_(k+1)to next time t_(k+2) for calculation is defined by the followingequation using state feedback gain (f₁, F₂).

    u(k+1)=-f.sub.1 X.sub.I (k+1)-F.sub.2 X(k+1)               (41)

The adder 107 supplies the state estimated value [X(k+1), X(k+1)]^(T) attime t_(k+1) to a commander 108 for heat generated by the heater. Thecommander 108 multiplies the state estimated value [X(k+1),X(k+1)]^(T)by the state feedback gain to define a command value of heat generatedby the heater.

(9) The above control calculation is executed after the next detectedvalue y(k+2) of film thickness is obtained from the sampler 100 at timet=t_(k+2) of calculation execution when the thickness gauge is movedalong the width of the film after the time period T and reaches theopposite film end.

(p) Example of Design

As a first actual example, an example of design is described in the casewhere transfer functions g₁ (s), g₂ (s) and g₃ (s) are given by thefollowing equations: ##EQU38## u_(i) (t)(i=1-5) is variation (watt) ofheat generated by the heater, and y_(i) (t)(i=1-5) is variation (micron)of thickness of film at the position of the thickness gaugecorresponding to the position of the heater. The dead time L₁ due tomovement of the film and times L₁ ' and L₂ " required for movement ofthe thickness gauge from the thickness control point 3' to the film endassume the following values.

    L.sub.1 =26 seconds

    L.sub.2 '=17 seconds

    L.sub.2 "=7.5 seconds.

Accordingly

    L.sub.B =43 seconds

    L.sub.C =33.5 seconds.

It is assumed that the thickness control point 3' exists at the end C ofthe film as shown in FIG. 8. The control calculation execution period Tassumes the following value.

    T=22.5 seconds                                             (72)

In order to design the control system, it is necessary to express therelation between the input u(t) and the output y(t) of the equation (1)and obtain the controllable and observable state equations (2) and (3).G(s) constituted of g₁ (s), g₂ (s) and g₃ (s) of the equations (69) to(71) can be expressed by an equation of the 77th degree, while thecontrollable and observable equation has been found to be an equation ofthe 29th degree. Accordingly, the equations (2) and (3) of the 29thdegree are obtained from G(s).

(1) Decision of State Feedback Gain Matrix F

The state feedback gain matrix F of the equation (11) is obtained as asolution of an optimum regulator problem for the state equation (8)extended to the equation of the 30th degree on the basis of the equation(2). Since the equation (8) is a state equation of a continuous timesystem, the equation is changed to a discrete state function with thesampling period T=22.5 seconds and a regulator solution is applied. Aproper estimation function is employed to obtain the state feedback gainmatrix F and as a result the following values are obtained as mainvalues for determining a response of control as the eigen values of thematrix (A-BF).

    0.856, 0.8119, 0.7755, 0.7618

Further, other eigen values except above are not described since theabsolute value thereof is small and attenuation is fast. Since all eigenvalues are within a circle having a radius of 1, stable control can beattained. Since the eigen value having the slowest attenuation is0.8560, the stabilization time Ts can be predicted as about 11 minutesfrom (0876)³⁰ ≈0.01 with definition of control error 1% as follows.

    Ts=T×35=22×30 sec.=675 sec.=11.3 min.

(2) Decision of Feedback Gain K of Operational Calculator

The feedback gain matrix K of the operational calculator of the equation(56) or (58) is obtained for the state equation (55) or (58) of the 29thdegree and the output equation (54) of the fifth degree. The gain matrixK is obtained as a solution of the optimum regulator problem so that thematrices {φ_(B) ^(T) -(Cφ_(B))^(T) K_(B) ^(T) } and {φ_(C) ^(T)-(Cφ_(C))^(T) K_(C) ^(T) } have a stable eigen value. For example, thediscrete time of the state equation (55) defining the gain matrix K_(B)is (T-L_(B) +L_(C))=(22.5-43+33.5)=13 seconds. As a result of obtainingthe gain matrix K using a proper estimation function, the followingvalues are obtained as main values for determining convergence of theoperational calculator as eigen values of the matrix (φ_(B) -K_(B)Cφ_(B)).

    0.9183, 0.9183, 0.9183, 0.9183, 0.9183, 0.7654, 0.7654, 0.7654, 0.7654, 0.7654,

Other eigen values except above are not described since the absolutevalues are small and convergence is fast. Since all the values arewithin a circle having a radius of 1, the estimated error can be reducedwith the lapse of time. Since the eigen value having the slowestattenuation is 0.9183, the time To required for attenuation of theestimated error to an initial 1% can be predicted from (0.9183)⁵⁵ ≅0.01as follows.

    To=(T-L.sub.B +L.sub.C)×55=13×55 sec.=715 sec.=12 min.

The gain matrix K_(C) having the stabilization time To of 12 minutes isobtained for the matrix φ_(C).

(q) Simulation Example 1

FIG. 20 shows an example of simulation result obtained by calculationusing the gain matrices F, K_(B) and K_(C) obtained above. FIG. 20(a)shows variations (variations of detected values of the thickness gauge)of five thickness values Y₁ to Y₅ versus time when the set value ofthickness Y₃ is changed stepwise by 5 micron. FIG. 20(b) showsvariations of amounts u₁ to u₅ of heat generated by the heaters in thesame condition as FIG. 13(a).

Since calculation is made after the execution period of 22.5 seconds ofcalculation after the set value of thickness has been changed, variationof heat generated by the heater occurs after 22.5 seconds from change ofthe set value of thickness. An amount of heat generated by the heater ismaintained to the same value until 22.5 seconds elapse and the nextcalculation is made. The calculation is made on the basis of a newlydetected value of thickness after 22.5 seconds to change an amount ofheat generated by the heater. Accordingly, an amount of heat generatedby the heater changes stepwise as shown in FIG. 20(b).

On the other hand, variation of the detected thickness value is detectedafter the lapse of the dead time L of 33.5 seconds after the amount ofheat generated by the heater has been changed after the lapse of 22.5seconds from the change of the set value. That is, variation ofthickness is detected after the lapse of 22.5+33.5=56 seconds after theset value of thickness has been changed. Thickness Y₃ is exactly changedto a set value and the change is substantially symmetrical to thethickness Y₃. Variation of heat generated by the heater u₃ is largest,variations by the heaters u₁ and u₅ are largest next to the heater u₃,and variations of the heaters u₂ and u₄ are smallest. This reason isbecause interference of the heaters u₂ and u₄ to thickness y₃ isreduced. The stabilization time which is estimated by the eigen valuedetermined by the gain matrices F, K_(B) and K_(C) and is 12 minutes issupported by FIG. 20. There is no variation in heat generated by theheater at the steady state, since the equation (66) is employed tocompensate the dead time instead of the equations (64) and (65).

When the present control system is applied actually, the samecalculation equation as that applied to the thickness y₃ is applied toeach of thicknesses y₁, y₂, y₄, and y₅ and each command value of heatmay be produced as a sum of results of the calculation equations. Itwill be understood that the control system considerably reducesinfluence of the dead time since the time required for stabilizingvariation of thickness when heat generated by the heater is changedstepwise without control of heat is about 13 minutes.

(r) Simulation Example 2

The second actual example is now described with reference to FIG. 21,which shows a control result when external heat of 8 watts is applied tothe heater u₃. FIG. 21(a) shows variations of thickness values y₁ to y₅versus time, and FIG. 21(b) shows variations of heat u₁ to u₅ generatedby the heaters versus time. As shown in FIG. 21(a), although thethickness y₃ is once increased by the external heat of the heater u₃,the thickness y₃ is returned to the original set value by changing theamounts of heat generated by the heaters u₁ to u₅ and the stabilizationtime is about 12 minutes in the same manner as FIG. 20. It is understoodthat variation due to the external disturbance is exactly compensated byintroducing the integrator in the present control system.

The thickness values y₂ and y₄ are once increased by influence ofexternal heat through thermal conduction along the width of the die. Thethickness values y₁ and y₅ are also influenced similarly, although theinfluence is small as compared with y₂ and y₄. In order to cancel theinfluence of such external heat, reduction of heat generated by theheater u₃ is largest, reduction by the heaters u₁ and u₅ is largest nextto the heater u₃, and reduction by the heaters u₂ and u₄ is smallest.This is because the reduction in the heaters u₂ and u₄ does notinfluence thickness y₃ so much. When external heat is applied to theheater u₃, other thickness values y₁, y₂, y₄ and y₅ are also changed,although such interference effect can be canceled by applying the samecalculation equation as that for the thickness value y₃ to each of thethickness values y₁, y₂ , y₄ and y₅.

A3. Effects of First Invention

The present invention is configured as described above and accordinglyhas the following effects. The integrator which time-integrates adifference between a detected value of thickness of film at apredetermined position and a set value of thickness is introduced and anoutput of the integrator is fed back to compensate an amount of heatgenerated by the heater for external heat influencing thickness of thefilm so that thickness of the film can be always identical with the setvalue. Further, in order to avoid large phase delay due to the deadtime, the state estimated value at time t-L earlier than the currenttime t by the dead time L is obtained by the operational calculator andthe state estimated value at time t-L is time-integrated by the stateshifter and the state prediction device during the dead time L so thatthe state estimated value at the current time t can be obtained toremove deterioration of control performance due to the dead time.

B1. First Embodiment of Second Invention

(a) Basic Configuration

The first embodiment of the second invention is described with FIGS. 22to 27. In order to avoid duplication, detailed description for the sameconfiguration as a conventional apparatus is omitted.

FIG. 22 is a block diagram of a film thickness controller forcontrolling heater and corresponding to a conventional adjustingmechanism 12b (FIG. 28). An output of a thickness gauge 11b is connectedto a thickness data memory 110. An arrival end identification outputsignal d from the thickness gauge 11b is connected to a distributor 111and a basic control system 112-i (i=1-N). A plurality of outputs of thedistributor 111 are connected to their corresponding basic controlsystems 112-i, respectively. Each of outputs of the basic control system112-i is connected to each of their corresponding command memories 113-ifor heat generated by the heaters. Each of outputs of the commandmemories 113-i is connected to superposition adder 114. An output of theadder 114 is connected to an operation value memory 115. An output ofthe memory 115 is connected back to the basic control systems 112-i.

(b) Basic Control system

Operation of one operating terminal device of the adjusting mechanismfor die lips changes thickness of a portion of film corresponding to anadjacent operating terminal device thereto. However, since theinterference range thereof is limited, there is considered the basiccontrol system including operating terminal devices disposed around acertain operating terminal device and disposed corresponding to portionsof film of which thickness is changed by operation of the certainoperating terminal device. The basic control system can control onlythickness of a portion of film corresponding to the operating terminaldevice selected as a center to a predetermined value of thickness. Moreparticularly, the basic control system can maintain the thickness of aportion of film corresponding to the certain operating terminal deviceto the predetermined value of thickness by varying operation values ofnot only the certain operating terminal device but also adjacentoperating terminal devices. The basic control system takes small numberof the operating terminal devices and interference to thickness of filmbetween operating terminal devices into consideration. A control systemhaving small number of operating terminal devices and having thefollowing merits is hereinafter referred to as a basic control system.

(i) Stability of the control system can be ensured because of smallnumber of operating terminal devices, and the control system having ahigh-speed response can be designed.

(ii) The control system which can control thickness of a portion of filmcorresponding to a central operating terminal device to thepredetermined value to compensate external disturbance even if externaldisturbance is applied to the central operating terminal device as wellas the adjacent operating terminal device can be designed.

(iii) Since interference to thickness of film between operating terminaldevices is considered, the control system which can effectivelydistribute operation values to operating terminal devices includingadjacent operating terminal devices to change thickness of a portion offilm corresponding to the central operating terminal device can bedesigned. That is, variation of the operation value of the centraloperating terminal device is large, while variation of the operationvalue of the adjacent operating terminal device is smaller as influencethereof to thickness of film is smaller.

(c) Variation of Operation Value in Adjacent Operating Terminal deviceas External Disturbance

In order to control thickness of the film over the whole width thereofstably with a high-speed response, the above basic control system isapplied to each of operating terminal devices of the adjustingmechanism. Thus, the stability of thickness control of the whole film isensured as follows.

(i) In a basic control system i' for a certain operating terminal devicei, thickness of a portion of film corresponding to the operatingterminal device is ensured to be controlled to the predetermined valueeven if external disturbance is added to the adjacent operating terminaldevice. When a basic control system (i+1)' is applied to an operationunit i+1 adjacent to the operation unit i, thickness of a portion offilm corresponding to the operating terminal device i+1 is ensured to becontrolled to the predetermined value stably.

(ii) The basic control system i' applied to the operating terminaldevice i can consider the operation value command in the basic controlsystem (i+1)' applied to the operating terminal device i+1 as anexternal disturbance applied to the operating terminal device of thebasic control system i'.

As described in the above item (b), the basic control system can stablycontrol thickness of a portion of film corresponding to the operatingterminal device i to which the basic control system i' is applied tocompensate external disturbance even if external disturbance is added tothe operating terminal device in the basic control system. Accordingly,thickness of a portion of film corresponding to the operating terminaldevice i can be controlled stably evwen if another basic control systemis applied to the operating terminal device i+1.

(d) Dead Time

In order to minimize interference effect to film thickness due to mutualinterference of the operating terminal devices of the adjustingmechanism 12b to control thickness of film over the whole width thereof,there is considered a control system which updates operation valuecommands for a multiplicity of operating terminal devicessimultaneously. To this end, it is necessary to move a thickness gaugein reciprocating manner along width of film to obtain all data ofthickness along the width of film and perform calculation each time thethickness gauge reaches an end of film. In this case, the thicknessgauge requires time to reach an end of film after measured thickness ofa certain portion of film. This time is a dead time until thecalculation is actually started after thickness data has been obtained.Accordingly, the dead time from after an operation value in theoperating terminal device has been changed until thickness of filminfluenced by the change of the operation value has been detected as athickness data and the detected thickness data is employed to performcalculation is a sum of a dead time L₁ due to movement of film from thedie lips to the thickness gauge and the above mentioned dead time L₂.That is, the dead time L of the equation (3b) is expressed by

    L=L.sub.1 +L.sub.2                                         (4b)

The thickness gauge measures thickness of film while being moved inreciprocating manner along the width of film. Since the film is moved ata certain speed, the thickness gauge measures thickness of film along alocus as shown in FIG. 27. If a position of a portion of film havingthickness t3 is indicated by a point C in FIG. 27, the dead time L₂ dueto movement of the thickness gauge in the case where calculation is madeat an end A of film is expressed by a time L₂ ' of movement of thethickness gauge between the points A and C in FIG. 27.

On the other hand, when calculation is made at an end B of film, thedead time L₂ due to movement of the thickness gauge is expressed by atime L₂ " of movement of the thickness gauge between the points C' and Bin FIG. 27. As seen from FIG. 27, since the dead times L₂ ' and L₂ " aregenerally different from each other, the control system for controllingthickness t3 to a predetermined value is characterized in that the deadtime L is different depending on whether the calculation is made at theend A or B. Accordingly, the thickness gauge produces an arrival endidentification signal for identifying whether the thickness gaugereaches the end A or B.

(e) Transfer Function Matrix

A basic control system is considered and this basic control system hasfive heaters h1 to h5 as operating terminal devices which are controlledby the basic control system, the five heaters being disposed in alongitudinal direction of a slot formed between the die lips. The basiccontrol system 112-i can control thickness of a portion of filmcorresponding to a central heater h3 to a predetermined value even ifexternal disturbance is added to the heaters h1 to h5. The reason thatadjacent heaters h1, h2 and h4, h5 are taken into consideration inaddition of the central heater h3 is because there is interference thatheat generated by the heater h3 changes thicknesses t1, t2 and t4, t5 offilm corresponding to the heaters h1, h2 and h4, h5 and influence to theheaters disposed outside of the heaters h1 and h5 by heat generated bythe heater h3 is negligible. Accordingly, control object for designingthe basic control system is expressed by the transfer function matrixG(s) of the following equation (1b): ##STR2## where U₁ (s) to U₅ (s) areLaplace transformed values of heat U₁ (t) to U₅ (t) generated by theheaters h1 to h5, Y₁ (s) to Y₅ (s) are Laplace transformed values ofthicknesses y₁ (t) to y₅ (t) of portions corresponding to the heaters h1to h5, and g₁ (s) to g₃ (s) are transfer functions corresponding torespective inputs and outputs. For example, g₁ '(s) is a transferfunction which produces temporal variation of thickness t3 when only theheater t3 is changed. In the transfer function matrix G(s) of theequation (1b), non-diagonal terms express mutual interference tothickness between heaters.

(e) State Equation

In order to express the relation between the inputs Ui(s) and theoutputs Yi(s) (i=1-5) of the equation (1b), the following equationconvenient for design of the control system is employed.

    X(t)=Ax(t)+Bu(t)                                           (2b)

    y(t)=Cx(t-L)                                               (3b)

where X is a state vector, u is an input vector in which u(t)=[u₁ (t),u₂ (t), u₃ (t), u₄ (t), u₅ (t)]^(T) (where T expresses transposition), yis an output vector in which y(t)=[y₁ (t), y₂ (t), y₃ (t), y₄ (t), y₅(t)]^(T), L of the equation (3b) is the dead time.

The equations (2b) and (3b) are controllable and observable. Therelation of the input u(t) and the output y(t) is expressed as in FIG.23 from the equations (2b) and (3b). Double line of FIG. 23 indicates avector value.

(f) Basic Control System as Solution of State Equation

In the first embodiment of the second invention, the basic controlsystem as a solution of the state equation is the control systemdescribed in detail in the first embodiment of the first invention.

Description is made to the basic control system in which operationamounts of the five heaters h1 to h5 around the heater h3 influence theoutput y₃ of the thickness gauge corresponding to the heater h3.

The basic control system satisfies the following conditions.

(1) Thickness y3 (hereinafter yi(t) is described as yi) is controlled toa predetermined value with good response even if external disturbance isadded to the heaters h1 to h5.

(2) In order to control thickness y3, operation amounts are assigned tothe heaters so that variation of operation amount in the heater h3 islargest, variation in the heaters h2 and h4 is largest next to theheater h3, and variation in the heaters h1 and h5 is smallest.

The basic control system satisfying the above conditions can be realizedby the control system having the configuration shown in FIG. 24.

Operation of the basic control system of FIG. 24 is described. Thethickness gauge detects thickness while being moved in reciprocatingmanner along the width of film. When the gauge reaches the end A or B offilm, measurement of thickness of film along the width thereof iscompleted. At this time the calculation is performed and accordingly theexecution period T of the calculation is substantially equal to a timerequired for movement of the thickness gauge along the width and isconsidered to be constant. Accordingly, the basic control system is adiscrete time control system.

(g) Operation of Basic Control System

Operational procedure of the basic control system of FIG. 27 is asfollows:

(1) It is assumed that the thickness gauge 11b reaches the end A or B offilm at the discrete time t=t_(k+1). At this time, a vector consistingof detected values of thickness y(t_(k+1))=y(k+1) (y₁ (k+1)˜y₅ (k+1) isobtained through the thickness gauge 11b and sampler 100. At the sametime, the thickness gauge produces the arrival end identification signald indicative of the end which the gauge has reached.

(2) Only thickness y₃ (k+1) of a portion of film corresponding to theheater h3, of the film thickness detection vector y(k+1) is supplied toa subtracter 101 which produces thickness deviation ε(k+1)=r₃ (k+1)-y₃(k+1) between the thickness y₃ (k+1) and a set value r₃ (k+1).

(3) An integrator 102 is supplied with the thickness deviation ε(k+1)from the subtracter 101 and produces a time-integrated value X_(I) (k+1)of the thickness deviation. The integrator 102 serves as an externaldisturbance compensator to compensate the external disturbance varyingthickness y₃ by heat generated by the heater and to control thickness y₃to be identical with a set value.

(4) The operational calculator 103 is supplied with a past time sequencedata (herein u(k)) of heat generated by the heater stored in a memory104 and the film thickness detection value y(k+1) and produces anestimated value X(t_(k+1) -L)=ω(k+1) of state variable at time (t_(k+1)-L) before time t_(k+1) by the dead time L defined by the arrival endidentification signal d produced from the thickness gauge.

(5) A state shifter 105 is supplied with the output x_(I) (k+1) of theintegrator 102 and the output ω(k+1) of the operational calculator 103and multiplies them by a coefficient for shifting the state by the deadtime L defined by the arrival end identification signal d produced bythe thickness gauge to obtain a state estimated value at time t_(k+1).

(6) A state prediction device 106 produces state variations for theinputs u(k) from time (t_(k+1) -L) to time t_(t+1) which are suppliedfrom the memory 104 which stores the past time sequence data of heatgenerated by the heater by the dead time defined by the arrival endidentification signal d produced by the thickness gauge.

(7) An adder 107 is supplied with an output of the state shifter 105 andan output of the stage prediction device 106 and produces as theaddition result thereof a state estimated value at time t_(k+1).Although the operational calculator 103 can not obtain only the stateestimated value at time (t_(k+1) -L) due to the dead time L, the stateshifter 105 and the state prediction device 106 effect integrationoperation during the dead time L to obtain the state estimated value attime t_(k+1). Since the above operation (5), (6) and (7) can removeinfluence of the phase delay due to the dead time L, thickness controlwith good response can be effected while maintaining the stability ofthe control system.

(8) A heat commander 108 multiplies the state estimated value from theadder 107 by the feedback gain to produce a heat command value to theoperating terminal device 109. If the operation amount of the operatingterminal device 109 is changed, thickness of the film is changed throughthickness process 130

(9) The above calculation is made each time a new film thicknessdetection value y(k+2) is obtained by the sampler 100 when the thicknessgauge 11b reaches the opposite end of film at time t_(k+2) and thicknessdata along the whole width of the film is newly obtained through thedead time 131.

(h) Thickness Control by Combined Basic Control Systems

The application procedure obtained as described above is shown in FIG.25.

FIG. 25(a) illustrates the application of the basic control system (1)in order to control thickness y₃ to a predetermined value. The basiccontrol system (1) detects thicknesses y₁ to y₅ and defines commandvalues u₁.sup.(1) to u₅.sup.(1) of heat generated by the heaterscorresponding to the thicknesses y₁ to y₅.

FIG. 25(b) illustrates the application of the basic control system (2)in order to control thickness y₄ to a predetermined value. The basiccontrol system (2) detects thicknesses y₂ to y₆ and defines commandvalues u₂.sup.(2) to u₆.sup.(2) of heat generated by the heaterscorresponding to the thicknesses y₂ to y₆.

FIG. 25(c) illustrates the application of the basic control system (3)in order to control thickness y₅ to a predetermined value. The basiccontrol system (3) detects thicknesses y₃ to y₇ and defines commandvalues u₃.sup.(3) to u₇.sup.(3) of heat generated by the heaterscorresponding to the thicknesses y₃ to y₇.

FIG. 25(d) illustrates the application of the basic control system (4)in order to control thickness y₆ to a predetermined value. The basiccontrol system (4) detects thicknesses y₄ to y₈ and defines commandvalues u₄.sup.(4) to u₈.sup.(4) of heat generated by the heaterscorresponding to the thicknesses y₄ to y₈.

FIG. 25(e) illustrates the application of the basic control system (5)in order to control thickness y₇ to a predetermined value. The basiccontrol system (5) detects thicknesses y₅ to y₉ and defines commandvalues u₅.sup.(5) to u₉.sup.(5) of heat generated by the heaterscorresponding to the thicknesses y₅ to y₉.

The final command value u₅ for the heater h5, for example, is given bythe following equation from the above basic control systems (1) to (5).

    u.sub.5 =(u.sub.5.sup.(1) +u.sub.5.sup.(2) +u.sub.5.sup.(3) +u.sub.5.sup.(4) +u.sub.5.sup.(5))×1/5              (4b)

As described above, the command value of heat generated by one heater h5is defined by application of five basic control systems.

(i) Stability of Thickness Control by Combined Basic Control Systems

Referring to FIG. 25, description is now made to operation that thebasic control systems are successively applied to control thickness of aportion of film corresponding to each of the operating terminal devices,that is, the heaters to the predetermined value so that thicknesscontrol of the whole film is made stably with good response.

The basic control system (3) which controls thickness y₅ of a portion offilm corresponding to the heater u₅ to a predetermined value is taken asan example. Since the command value of heat generated by the heater h(3)is given by an averaged addition value (u₃.sup.(3) +u₃.sup.(1)+u₃.sup.(2))×1/3 of the command values u₃.sup.(3), u₃.sup.(1) andu₃.sup.(2) of the basic control systems (3), (1) and (2), respectively,it is considered that the heater h3 is influenced by a kind of externalheat of (u₃.sup.(1) +u₃.sup.(2))×1/3. Then, since the command value ofheat generated by the heater h4 is given by an averaged addition value(u₄.sup.(3) +u₄.sup.(1) +u₄.sup.(2) +u₄.sup.(4))×1/4 of the commandvalues u₄.sup.(3), u⁴(1), u₄.sup.(2) and u₄.sup.(4) of the basic controlsystems (3), (1), (2) and (4), respectively, it is considered that theheater h4 is influenced by external heat of (u₄.sup.(1) +u₄.sup.(2)+u₄.sup.(4))×1/4. Since the command value of heat generated by theheater h5 is given by an averaged addition value (u₅.sup.(3) +u₅.sup.(1)+u₅.sup.(2) +u₅.sup.(4) +u₅.sup.(5))×1/5 of the command valuesu₅.sup.(3), u₅.sup.(1), u₅.sup.(2), u₅.sup.(4), u₅.sup.(5) of the basiccontrol systems (3), (1), (2), (4) and (5), respectively, it isconsidered that the heater h5 is influenced by external heat of(u₅.sup.(1) +u₅.sup.(2) +u₅.sup.(4) +u₅.sup.(5))×1/5. The command valueof heat generated by the heater h6 is considered to be influenced byexternal heat having an averaged addition value (u₆.sup.(3) +u₆.sup.(2)+u₆.sup.(4) +u₆.sup.(5))×1/4 of the command values u₆.sup.(3), u⁶(2),u₆.sup. (4) and u₆.sup.(5) of the basic control systems (3), (2), (4)and (5), respectively. Finally, since the command value of heatgenerated by the heater h7 is given by an averaged addition value(u₇.sup.(3) +u₇.sup.(4) +u₇.sup.(5))×1/3 of the command valuesu₇.sup.(3), u⁷(4) and u₇.sup.(5) of the basic control systems (3), (4)and (5), respectively, it is considered that the heater h7 is influencedby external heat of (u₇.sup.(4) +u₇.sup.(5))×1/3.

As described above, it is considered that all of the heaters of thebasic control system (3) are influenced by external heat from theadjacent control systems. However, since the basic control systems (3)can control thickness y₅ to the predetermined value as described aboveeven if external heat is added to the heaters 3 to 7, it is understoodthat control by the control basic device (3) to thickness y₅ is madestably. This can be applied to the other basic control systems whichcontrol thickness of other portions and accordingly it is understoodthat thickness control is stably made over the whole film.

(j) Configuration and Operation of Second Invention

Configuration of the second invention is described with reference toFIG. 22.

Since the thickness gauge 11b is moved in reciprocating manner along thewidth of film to detect thickness of film, thickness data over the widthof film is obtained each time the thickness gauge reaches the end offilm. The thickness data over the width of film is supplied to thethickness data memory 110.

On the other hand, the thickness gauge 11b supplies the arrival endidentification signal indicative of the end which the thickness gaugehas reaches to the distributor 111 and the basic control systems 112-i(i=1-N) each time the thickness gauge has reached the end of film. Whenthe distributor 111 is supplied with the arrival end identificationsignal from the thickness gauge 11b, the distributor 111 reads out a setof thickness data necessary for the basic control systems 112-i from thethickness data memory 110 and supplies the set of thickness data to thepredetermined basic control systems 112-i. Accordingly, the set ofthickness data is simultaneously distributed to to the basic controlsystems which control thickness of portions of film corresponding to theheaters in synchronism with the arrival end identification signal. Thebasic control systems 112-i is supplied with the set of thickness datafrom the distributor 111 and data of the operation value memory andidentifies the end of film which the thickness gauge has reached on thebasis of the arrival end identification signal to select the correctdead time L and execute calculation so that a predetermined number ofcommand values of heat are stored in the command value memories 113-2 to113-N. When the command value memories 113-1 to 113-N are supplied withthe command values of heat from all of the basic control systems 112-1to 112-N, the superposition adder 114 adds outputs of the command valuememories 113-1 to 113-N for each heater and calculates an average valuethereof to define a final command value S of heat for each heater.

The command value S of the superposition adder 114 is stored in theoperation value memory 115. Then, when the thickness gauge 11b has beenmoved and reached the opposite end of film so that a new arrival endidentification signal has been produced, the distributor 111, the basiccontrol systems 112-i (i=1-N) and the superposition adder 114 are alloperated as described above so that all command values of heat areupdated.

As described above, the basic control systems can control thickness ofportions of film corresponding to the heaters to a predetermined valueover the width of film stably.

(k) Example of Design

An example of design is described in the case where transfer functionsg₁ (s), g₂ (s) and g₃ (s) are given by the following equations:##EQU39## The basic control systems (1) to (6) as shown in FIG. 25, tenheaters h1 to h10, and ten points t1 to t10 of thickness correspondingto positions of the heaters are assumed and it is considered thatthicknesses y₃ to y₈ are controlled to a predetermined value. u_(i)(t)(i=1-10) is variation (Kcal/s) of heat generated by the heater, andy_(i) (t)(i=1-10) is variation (cm) of thickness of film at the positionof the thickness gauge corresponding to the position of the heater. Thedead time L₁ due to movement of the film and times L₂ ' and L₂ "(referred to FIG. 27) required for movement of the thickness gauge fromthe thickness control point 3 to 8 to the film end assume a value of thefollowing equation and values shown in Table 1.

    L.sub.1 =30 seconds

                  TABLE 1                                                         ______________________________________                                        Dead Time L at Thickness Control Points                                       Thickness                                                                     Control                                                                       Point    3       4       5     6     7     8                                  ______________________________________                                        Dead Time                                                                              1.5     2.25    3.0   3.75  4.5   5.25                               L.sub.2 " (sec)                                                               Dead Time                                                                              15      14.25   13.5  12.75 12    11.25                              L.sub.2 " (sec)                                                               Whole Dead                                                                             31.5    32.5    33.0  33.7  34.5  35.25                              Time L of                                                                     End (A) (sec)                                                                 (L.sub.1 + L.sub.2)                                                           The same of                                                                            45.0    44.25   43.5  42.75 42.0  41.25                              End (B)                                                                       ______________________________________                                    

It is assumed that the thickness control point 3 exists at the end A ofthe film as shown in FIG. 27. The control calculation execution period Tassumes the following value.

    T=16.5 seconds

In order to design the control system, it is necessary to express therelation between the input u(t) and the output y(t) of the equation (1b)and obtain the controllable and observable state equations (2b) and(3b). G(s) constituted of g₁ (s), g₂ (s) and g₃ (s) of the equations(5b) to (7b) can be expressed by an equation of the 77th degree, whilethe controllable and observable equation has been found to be anequation of the 39th degree. Accordingly, the equations (2b) and (3b) ofthe 39th degree are obtained from G(s).

(1) Decision of State Feedback Gain Matrix

The state feedback gain matrix of the basic control system is obtainedas a solution of an optimum regulator problem for the state equationextended to the equation of the 40th degree by introducing theintegrator for compensation of external disturbance on the basis of theequation (2b). Since the calculation is made every T=16.5 seconds, thestate equation of the continuous time system is changed to a discretestate function with the sampling period T=16.5 seconds and a regulatorsolution is applied. A proper estimation function is employed to obtainthe state feedback gain matrix and as a result the following values areobtained as the eigen values of the control system.

    0.876±0.02i, 0.79, 0.50±0.07i, 0.60±0.09i, 0.60±0.06i, 0.51

Further, 30 eigen values other than above are not described since theabsolute value thereof is less than 0.1 and attenuation is fast. Sinceall eigen values are within a circle having a radius of 1, stablecontrol can be attained. Since the eigen value having the slowestattenuation is 0.88±0.02i, the stabilization time Ts can be predicted asabout 10 minutes from (0876)³⁵ ≈0.01 with definition of control error 1%as follows.

    Ts=T×35=16.5×35 sec.=577.5 sec.=9.6 min.

(2) Decision of Feedback Gain of Operational Calculator

The feedback gain matrix of the operational calculator which estimatesthe state before time t_(k+1) for calculation execution by the dead timeL is obtained for the state equation of the 39th degree and the outputequation of the fifth degree. The gain matrix K is obtained as asolution of the optimum regulator problem using a proper estimationfunction. The following values are obtained as eigen values of theoperational calculator for the obtained gain matrix.

    0.9077±0.0002i, 0.9076, 0.9075, 0.9075, 0.772±0.0001i, 0.722, 0.722, 0.722, 0.576±1×10.sup.-5 i, 0.576±1×10.sup.-5 i, 0.232, 0.232, 0.232, 0.232, 0.232

20 eigen values other than above concentrate to the origin. Since allthe values are within a circle having a radius of 1, the estimated errorcan be reduced with the lapse of time. Since the eigen value having theslowest attenuation is 0.9077, the time To required for attenuation ofthe estimated error to an initial 1% can be predicted from (0.9077)⁴⁵≈0.01 as follows.

    To=T×45=16.5×45 sec.=742.5 sec.=12.4 min.

(1) Simulation Example 1

FIGS. 31 and 32 show an example of simulation result obtained bycalculation using the state feedback and the gain of the operationalcalculation obtained above.

FIGS. 31 and 32 show variations of thickness and variation of heatgenerated by the heaters when the set values of thickness y₃ to y₈ arechanged stepwise by 0.02 mm. FIG. 31(a) shows variations of five amountsy₁ to y₅ of thickness (variation of the detected value of the thicknessgauge) versus time. FIG. 31(b) shows variations of heat U₁ to U₅generated by the heaters at this time in the same manner as FIG. 31(a).FIG. 32(a) shows variations of thickness y₆ to y₁₀ and FIG. 32(b) showsvariations of heat U₆ to U₁₀ generated by the heater.

Since calculation is made after the execution period of 16.5 seconds ofcalculation after the set value of thickness has been changed, variationof heat generated by the heater occurs after 16.5 seconds from change ofthe set value of thickness. An amount of heat generated by the heater ismaintained to the same value until 16.5 seconds elapse and the nextcalculation is made. The calculation is made on the basis of a newlydetected value of thickness after 16.5 seconds to change an amount ofheat generated by the heater. Accordingly, an amount of heat generatedby the heater changes stepwise as shown in FIG. 31 and 32(b).

On the other hand, variation of the detected thickness value is detectedafter the lapse of the dead time L after the amount of heat generated bythe heater has been changed after the lapse of 16.5 seconds from thechange of the set value. For example, when calculation is made withthickness y₃ for the end A shown in FIG. 27, the dead time L is 31.5seconds from Table 1. That is, variation of thickness is detected afterthe lapse of 16.5+31.5=48 seconds after the set value of thickness hasbeen changed. Thickness y₃ is exactly changed to a set value as can beseen from FIGS. 31 and 32. The heaters h1, h2, h9 and h10 are introducedin consideration of mutual interference to thicknesses y₃ and y₈ and thethicknesses y₁, y₂, y₉ and y₁₀ corresponding to the heaters h1, h2, h9and h10 are not controlled to the set value. On the other hand,variations of heat generated by the heaters U₃ and U₈ at the end in thethickness control region are largest, variations by the heaters U₄ to U₇located in the center are largest next to the heaters U₃ and U₈, andvariations of the heaters U₁, U₂, U₉ and U₁₀ located outside of thecontrol region are smallest.

The stabilization time is about 18.5 minutes which is considerably largeas compared with the stabilization time of 12.4 minutes calculated bythe eigen value of the operational calculator (the stabilization time bythe eigen value of the regulator is still shorter). This is based on thefollowing reason.

In order to prevent the command value of heat generated by the heaterfrom being changed largely for each calculation, the command value isdefined with weight added as follows.

    U.sub.d.k =WU.sub.d.k-1 +(1-W)U.sub.k                      (8b)

where

U_(d).k : command value of heat defined by the calculation time t=t_(k),

U_(d).k-1 : command value of heat defined by the last calculation timet=t_(k-1),

U_(k) : command value of heat calculated at the calculation timet=t_(k), and

W: weight coefficient.

In this simulation, W=0.8. This means that when the calculation periodT=16.5 seconds is considered, a time delay corresponding to a delay offirst order having a time constant of 74.65 seconds is added to the heatcommander. Accordingly, it is considered that the stabilization time ofthickness control of FIGS. 31, 32 is larger than the stabilization timeestimated by the eigen value of the operational calculator. Then, evenif the thickness control is in the stabilization state, the commandvalue of heat changes for each calculation. The reason is because themagnitude of the dead time L in the calculation in the state shifter ofthe basic control system is different in one end A and the other end Bof the film for calculation.

(m) Simulation 2

FIGS. 33 and 34 shows a control result when external heat of 8.4 wattageis applied to the heater U₃ to U₈. FIG. 33(a) shows variations ofthickness values y₁ to y₅ versus time, and FIG. 33(b) shows variationsof heat U₁ to U₅ generated by the heaters versus time. FIG. 34(a) showsvariations of thickness values y₆ to y₁₀ versus time and FIG. 34(b)shows variations of heat U₆ to U₁₀ generated by the heaters versus time.

As seen in FIGS. 33 and 34(a), although the thickness values y₃ to y₈are once increased by the external heat of the heater U₃ to U₈, thethickness values y₃ to y₈ are returned to the original set value bychanging the amounts of heat generated by the heaters U₁ to U₁₀ and thestabilization time is about 18.5 minutes in the same manner as FIGS. 31and 33. It is understood that variation due to the external disturbanceis exactly compensated by introducing the integrator in the presentcontrol system. The thickness values y₁, y₂, y₉ and y₁₀ are onceincreased by influence of external heat through thermal conduction alongthe width of the die. In order to cancel the influence of such externalheat, reductions of amounts U₃ to U₈ of heat generated by the heaterlocated outside of the control region are largest, and reductions ofamounts U₁, U₂, U₉ and U₁₀ generated by the heaters located outside ofthe control region is smallest.

B2. Second Embodiment of Second Invention

(a) Relation to First Embodiment of Second Invention

The first embodiment of the second invention employs the control systemsof the first embodiment of the first invention as the basic controlsystems, while the second embodiment of the second invention employs thecontrol systems of the second embodiment of the first invention as basiccontrol systems.

(b) Dead Time

The thickness gauge measures thickness of film along a locus as shown inFIG. 27. If a position of thickness t₃ is indicated by the point C inFIG. 27, the dead time L₂ due to movement of the thickness gauge in thecase where calculation is made at the end A of film is expressed by atime L₂ ' corresponding to movement between the points C and A of FIG.27.

On the other hand, when calculation is made at the end B of film, thedead time L₂ due to movement of the thickness gauge is expressed by atime L₂ " corresponding to movement between the points C' and B in FIG.27. As can be seen from FIG. 27, since the dead time L₂ ' is generallydifferent from the dead time L₂ ", the control system which controlsthickness t₃ to a predetermined value is characterized in that the deadtime 1 of the equation (3b) is different depending on whethercalculation is made at the end A or B of film. That is: the dead timeL_(A) for the end A is given by

    L.sub.A =L.sub.1 +L.sub.2 '                                (9b)

the dead time L_(B) for the end B is given by

    L.sub.B =L.sub.1 +L.sub.2 "                                (10b)

Accordingly, the thickness gauge produces an arrival end identificationsignal for identifying the end A or B which the thickness gauge hasreached.

The thickness gauge is moved in reciprocating manner along the width offilm as shown in FIG. 27 to detect thickness of film and finishesmeasurement of thickness over the width of film when the thickness gaugehas reached the end A or B of film. At this time, the calculation isexecuted and accordingly the execution period of calculation issubstantially equal to a time required for movement of the thicknessgauge over the width of film and the period is considered to beconstant. Thus, the basic control system is a discrete time controlsystem.

(c) Basic Control System

The state equations (2b) and (3b) are controllable and observable. Therelation of the input u(t) and the output y(t) is shown in FIG. 23 fromthe equations (2b) and (3b). Double line in FIG. 23 indicates vectorvalue. A configuration of the basic control system of the secondembodiment is also the same as that of FIG. 24. Double line of FIG. 24indicates vector value. The configuration of the basic control systemshown in FIG. 24 is as follows:

(1) It is assumed that the thickness gauge 11b reaches the end A or B offilm at the discrete time t=t_(k+1). At this time, a vector consistingof detected values of thickness y(t_(k+1))=y(k+1)(y₁ (k+1)˜y₅ (k+1) isobtained through the thickness gauge 11b and sampler 100. At the sametime, the thickness gauge produces the arrival end identification signald indicative of the end which the gauge has reached.

(2) Only thickness y₃ (k+1) of a portion of film corresponding to theheater h3, of the film thickness detection vector y(k+1) is supplied toa subtracter 101 which produces thickness deviation ε(k+1)=r₃ (k+1)-y₃(k+1) between the thickness y₃ (k+1) and a set value r₃ (k+1).

(3) An integrator 102 is supplied with the thickness deviation ε(k+1)from the subtracter 101 and produces a time-integrated value X_(I) (k+1)of the thickness deviation. The integrator 102 serves as an externaldisturbance compensator to compensate the external disturbance varyingthickness y₃ by heat generated by the heater and to control thickness y₃to be identical with a set value.

(4) The operational calculator 103 is supplied with a past time sequencedata (herein u(k)) of heat generated by the heater stored in a memory104 and the film thickness detection value y(k+1) and produces anestimated value X(t_(k+1) -L)=ω(k+1) of state variable at time (t_(k+1)-L) before time t_(k+1) by the dead time L defined by the arrival endidentification signal d produced from the thickness gauge.

(5) A state shifter 105 is supplied with the output x_(I) (k+1) of theintegrator 102 and the output ω(k+1) of the operational calculator 103and multiplies them by a coefficient for shifting the state by theaverage dead time L which is an average value of the dead time L_(A)(refer to the equation (9b)) in the case where the thickness gauge hasreached the end A and the dead time L_(B) (refer to the equation (19b))in the case where the thickness gauge has reached the end B to obtain astate estimated value at time t_(k+1).

    L=(L.sub.A +L.sub.B)/2                                     (11b)

From the equations (9b), (10b) and (11b), the dead time L is given by

    L=L.sub.1 +(L.sub.2 '+L.sub.2 ")/2                         (12b)

(L₂ '+L₂ ") is substantially equal to a time required for movement ofthe thickness gauge over the width of film and accordingly is equal tothe execution period t of calculation. Thus, from the equation (12b),the average dead time L is given by

    L=L.sub.1 +T/2                                             (13b)

As seen from the equation (13b), the average dead time L is constantirrespective of the end of film which the thickness gauge reaches.

(6) A state prediction device 106 produces state variations for theinputs u(k) from time (t_(k+1) -L) to time t_(t+1) which are suppliedfrom the memory 104 which stores the past time sequence data of heatgenerated by the heater by the average dead time in the same manner asthe state shifter 105.

(7) An adder 107 is supplied with an output of the state shifter 105 andan output of the stage prediction device 106 and produces as theaddition result thereof a state estimated value at time t_(k+1).Although the operational calculator 103 can not obtain only the stateestimated value at time (t_(k+1) -L) due to the dead time L, the stateshifter 105 and the state prediction device 106 effect integrationoperation during the average dead time L to obtain the state estimatedvalue at time t_(k+1). Since the above operation (5), (6) and (7) canremove influence the phase delay due to the average dead time L,thickness control with good response can be effected while maintainingthe stability of the control system.

(8) A heat commander 108 multiplies the state estimated value from theadder 107 by the feedback gain to produce a heat command value to theoperating terminal device 109. If the operation amount of the operatingterminal device 109 is changed, thickness of the film is changed throughthickness process 130

(9) The above calculation is made each time a new film thicknessdetection value y(k+2) is obtained by the sampler 100 when the thicknessgauge 11b reaches the opposite end of film at time t_(k+2) and thicknessdata along the whole width of the film is newly obtained through thedead time 131.

(d) Average Dead Time

The reason that the average dead time L is used as the integration timein the state shifter 105 and the state prediction device 106 instead ofthe dead times L_(A) and L_(B) is now described.

If the integration section corresponding to the dead time L_(A) or L_(B)different from each other by the calculation for the end A or B isassumed, the state estimated value at time t_(k+1) is not continuous foreach calculation and changes stepwise. When the dead time L_(A) islarger than the dead time L_(B), the state estimated value at the end Ais larger than the state estimated value at the end B and the operationvalue of the heater defined by multiplying the state estimated value bythe feedback gain is also repeatedly varied unevenly. There is adrawback that variation of the operation value is maintained even in thesteady state. On the other hand, if the average dead time L is used forthe calculation at the ends A and B in common, there is no state inwhich the state estimated value is incontinuous at the ends A and Bbecause of the identical integration section and uneven variation of theoperation value in the steady state is removed.

(e) Thickness Control by Combined Basic Control Systems

The first embodiment of the second invention is identical with thesecond embodiment thereof with the exception that only the basic controlsystems are different. Combination of the basic control systems is thesame. Accordingly, description for thickness control by the combinedbasic control systems in the first embodiment of the second inventioncan be all applied to the second embodiment. That is, description inB1(h) to (j) is all applied to B2.

(f) Design Example

An actual example is now described. As a first actual example, anexample of design is described in the case where transfer functions g₁(s), g₂ (s) and g₃ (s) are given by the following equations: ##EQU40##The basic control systems (1) to (6) as shown in FIG. 25, ten heaters h1to h10, and ten points t1 to t10 of thickness corresponding to positionsof the heaters are assumed and it is considered that thicknesses y₃ toy₈ are controlled to a predetermined value. U_(i) (t)(i=1-10) isvariation (watt) of heat generated by the heater, and y_(i) (t)(i=1-10)is variation (micron) of thickness of film at the position of thethickness gauge corresponding to the position of the heater. The deadtime L₁ due to movement of the film and times L₂ ' and L₂ " (referred toFIG. 27) required for movement of the thickness gauge from the thicknesscontrol point 3 to 8 to the film end assume a value of the followingequation and values shown in Table 2.

    L.sub.1 =26 seconds

                  TABLE 2                                                         ______________________________________                                        Dead Time L at Thickness Control Points                                       Thickness                                                                     Control                                                                       Point    3       4       5     6     7     8                                  ______________________________________                                        Dead Time                                                                              2.8     3.75    4.7   5.6   6.6   7.5                                L.sub.2 ' (sec)                                                               Dead Time                                                                              19.7    18.75   17.8  16.9  15.9  15.0                               L.sub.2 " (sec)                                                               Whole Dead                                                                             28.8    29.75   30.7  31.6  32.6  33.5                               Time L of                                                                     End (A) (sec)                                                                 (L.sub.1 + L.sub.2)                                                           The same of                                                                            45.7    44.75   43.8  42.9  41.9  41.0                               End (B)                                                                       ______________________________________                                    

It is assumed that the thickness control point 3 exists at the end A ofthe film as shown in FIG. 27. The control calculation execution period Tassumes the following value.

    T=22.5 seconds

In order to design the control system, it is necessary to express therelation between the input u(t) and the output y(t) of the equation (1b)and obtain the controllable and observable state equations (2b) and(3b). G(s) constituted of g₁ (s), g₂ (s) and g₃ (s) of the equations(14b) to (16b) can be expressed by an equation of the 77th degree, whilethe controllable and observable equation has been found to be anequation of the 29th degree. Accordingly, the equations (2b) and (3b) ofthe 29th degree are obtained from G(s).

(1) Decision of State Feedback Gain Matrix

The state feedback gain matrix of the basic control system is obtainedas a solution of an optimum regulator problem for the state equationextended to the equation of the 30th degree by introducing theintegrator for compensation of external disturbance on the basis of theequation (2b). Since the calculation is made every T=22.5 seconds, thestate equation of the continuous time system is changed to a discretestate function with the sampling period T=22.5 seconds and a regulatorsolution is applied. A proper estimation function is employed to obtainthe state feedback gain matrix and as a result the following values areobtained as main values for determining the response of the controlsystem as the eigen values of the control system.

    0.856, 0.8119, 0.7755, 0.7618

Further, eigen values other than above are not described since theabsolute value thereof is small and attenuation is fast. Since all eigenvalues are within a circle having a radius of 1, stable control can beattained. Since the eigen value having the slowest attenuation is 0.856,the stabilization time Ts can be predicted as about 12 minutes from(0.856)³⁰ ≈0.01 with definition of control error 1% as follows.

    Ts=T×30=22.5×30 sec.=675 sec.=11.3 min.

(2) Decision of Feedback Gain of Operational Calculator

The feedback gain matrix of the operational calculator which estimatesthe state before time t_(k+1) for calculation execution by the dead timeL is obtained for the state equation of the 29th degree and the outputequation of the fifth degree. FIG. 35 is a diagram illustrating thediscrete time used to transforms the state equation (2b) to the discreteequation in order to obtain the gain matrix of the operationalcalculation. In FIG. 35, it is assumed that the estimated value X(t_(k)-L_(B)) of the state variable at the past time by the dead time L_(B)has been already obtained in the calculation at the end B performed attime t_(k). In order to obtain the estimated value X(t_(k+1) -L_(A)) ofthe state variable at the past time by the dead time L_(A) in thecalculation at the end A performed at time t_(k+1), the state equation(2b) must be transformed to a discrete form with a time difference(t_(k+1) -L_(A))-(t_(k-) L_(B))=t_(k+1) -t_(k) -L_(A) +L_(B). Thediscrete time is (T-L_(A) +L_(B)) because of t_(k+1) -t_(k) =T. Thediscrete time (T-L_(A) +L_(B)) for the thickness control point 3 iscalculated from L_(A) =28.8 seconds and L_(B) =45.7 seconds in Table 2as follows:

    T-L.sub.A +L.sub.B =39.4 seconds

For the state equation transformed to the discrete form with 39.4seconds, a proper evaluation function is employed to obtain the gainmatrix of the operational calculation as a solution of an optimumregulator problem. The following values are obtained as main values fordetermining convergence of the operational calculation as eigen valuesof the operational calculator for the obtained gain matrix.

    0.7743, 0.7743, 0.7743, 0.7743, 0.7743, 0.4484, 0.4484, 0.4484, 0.4484, 0.4484

Since eigen values other than above are small and convergence is fast,they are not described. Since all the values are within a circle havinga radius of 1, the estimated error can be reduced with lapse of time.Since the eigen value having the slowest attenuation is 0.7743, the timeTo required for attenuation of the estimated error to an initial 1% canbe predicted from (0.7743)¹⁸ ≈0.01 as follows. ##EQU41## For otherthickness control points, the gain matrix of the operational calculationhaving the stabilization time To of 12 minutes was obtained in the samemanner.

(g) Simulation 1

FIGS. 36 and 37 show an example of simulation result obtained bycalculation using the state feedback and the gain of the operationalcalculation obtained above.

FIGS. 36 and 37 show variations of thickness and variation of heatgenerated by the heaters when the set values of thickness y₃ to y₈ arechanged stepwise by 5 micron. FIG. 36(a) shows variations of fiveamounts y₁ to y₅ of thickness (variation of the detected value of thethickness gauge) versus time. FIG. 36(b) shows variations of heat u₁ tou₅ generated by the heaters at this time in the same manner as FIG.36(a). FIG. 37(a) shows variations of thickness y₆ to y₁₀ and FIG. 37(b)shows variations of heat u₆ to u₁₀ generated by the heater.

Since calculation is made after the execution period of 22.5 seconds ofcalculation after the set value of thickness has been changed, variationof heat generated by the heater occurs after 22.5 seconds from change ofthe set value of thickness. An amount of heat generated by the heater ismaintained to the same value until 22.5 seconds elapse and the nextcalculation is made. The calculation is made on the basis of a newlydetected value of thickness after 22.5 seconds to change an amount ofheat generated by the heater. Accordingly, an amount of heat generatedby the heater changes stepwise as shown in FIG. 36 and 37(b).

On the other hand, variation of the detected thickness value is detectedafter the lapse of the dead time L after the amount of heat generated bythe heater has been changed after the lapse of 22.5 seconds from thechange of the set value. For example, when calculation is made withthickness y₃ for the end A shown in FIG. 37, the dead time L is 28.8seconds from Table 2. That is, variation of thickness is detected afterthe lapse of 22.5+28.8=51.3 seconds after the set value of thickness hasbeen changed. Thickness y₃ is exactly changed to a set value as can beseen from FIGS. 36 and 37. The heaters h1, h2, h9 and h10 are introducedin consideration of mutual interference to thicknesses y₃ and y₈ and thethicknesses y₁, y₂, y₉ and y₁₀ corresponding to the heaters h1, h2, h9and h10 are not controlled to the set value. On the other hand,variations of heat generated by the heaters u₃ and u₈ at the end in thethickness control region are largest, variations by the heaters u₄ to u₇located in the center are largest next to the heaters u₃ and u₈, andvariations of the heaters u₁, u₂, u₉ and u₁₀ located outside of thecontrol region are smallest.

As can be seen from FIGS. 36 and 37, thickness is controlled to thepredetermined value in about 12 minutes after a set value of thicknesshas been changed, that is, the stabilization time 12 minutes supports aresult estimated from the above mentioned eigen value.

(h) Simulation 2

A second actual example is now described with reference to FIGS. 38 and39.

FIGS. 38 and 39 shows a control result when external heat of 8 wattageis applied to the heater u₃ to u₈. FIG. 38(a) shows variations ofthickness values y₁ to y₅ versus time, and FIG. 38(b) shows variationsof heat u₁ to u₅ generated by the heaters versus time. FIG. 39(a) showsvariations of thickness values y₆ to y₁₀ versus time and FIG. 39(b)shows variations of heat u₆ to u₁₀ generated by the heaters versus time.

As seen in FIGS. 38 and 39(a), although the thickness values y₃ to y₈are once increased by the external heat of the heater u₃ to u₈, thethickness values y₃ to y₈ are returned to the original set value bychanging the amounts of heat generated by the heaters u₁ to u₁₀ and thestabilization time is about 12 minutes in the same manner as FIGS. 36and 37. It is understood that variation due to the external disturbanceis exactly compensated by introducing the integrator in the presentcontrol system. The thickness values y₁, y₂, y₉ and y₁₀ are onceincreased by influence of external heat through thermal conduction alongthe width of the die. In order to cancel the influence of such externalheat, reductions of amounts u₃ to u₈ of heat generated by the heaterlocated outside of the control region are largest, and reductions ofamounts u₁, u₂, u₉ and u₁₀ generated by the heaters located outside ofthe control region is smallest.

B3. Effects of Second Invention

As described above, according to the second invention, the adjustingmechanism for controlling thickness of film includes the die providedwith a multiplicity of operating terminal devices disposed along thewidth of film so that thickness control of a portion of filmcorresponding to one operating terminal device is effected to compensateexternal distrubance added to the operating terminal device and itsadjacent terminal devices, and there is provided the state predictionfunction to remove influence due to the dead time for thicknessdetection so that the basic control systems with good response can beapplied to control thickness of film to the predetermined value.Further, the basic control system is applied for each control ofthickness of a portion of film corresponding to the operating terminaldevice so that thickness control over the whole width of film isperformed stably.

I claim:
 1. A film thickness controller for use in an extrusion moldingapparatus and a flowing type molding apparatus including a die having aslot along which a plurality of operating terminal devices of adischarge amount adjusting mechanism of molten plastic are disposed anda thickness gauge for detecting variation of thickness after the lapseof a dead time corresponding to a time required for movement of the filmbetween the die and the thickness gauge, comprising a thickness datamemory for storing thickness data measured by said thickness gauge, adistributor for receiving an output of said thickness data memory and anarrival end identification signal which is produced by the thicknessgauge to identify whether the thickness gauge reaches either of bothends of the film, a plurality of basic control means for receiving anoutput of said distributor and the arrival end identification signalproduced by the thickness gauge, a plurality of command value memorieseach receiving an output of each of said plurality of basic controlmeans, a superposition adder for receiving an output of each of saidcommand value memories, and an operation value memory for receiving anoutput of said superposition adder and for supplying an output of saidoperation value memory to said plurality of basic control means.
 2. Afilm thickness controller according to claim 1, wherein:the thicknessgauge in the extrusion molding apparatus forms means for moving in areciprocating manner along the width of film to detect thickness of filmand to obtain thickness data over the width of film each time thethickness gauge reaches the end of film and for supplying the thicknessdata over the width of film; to the thickness data memory; saidthickness gauge further constituting means for furnishing the arrivalend identification signal to indicate the end which the thickness gaugehas reached to the distributor and the basic control systems each timethe thickness gauge has reached the end of film; said distributorconstituting means responsive to the arrival of an end identificationsignal from the thickness gauge for reading out a set of thickness datanecessary for the basic control systems from the thickness data memoryand for supplying the set of thickness data to the predetermined basiccontrol systems; said basic control means are for receiving the set ofthickness data from the distributor and data of the operation valuememory and for identifying the end of film which the thickness gauge hasreached on the basis of the arrival end identification signal to selecta correct dead time; said control means being further for calculating apredetermined number of command values of heat and storing the values inthe command value memories; said superposition adder constituting meansfor responding to supply to the command value memories of the commandvalues of heat from said basic control systems and for adding outputs ofthe command value memories for each heater; said superposition adderfurther constituting means for calculating an average value to define afinal command value S of heat for each heater; said operation valuememory constituting means for storing the command value S of thesuperposition adder; said distributor constituting means for respondingto the thickness gauge reaching the opposite end of film; saiddistributor further consituting means for producing a new arrival endidentification signal; to operate the basic control systems and thesuperposition adder again as to update all command values of heat.
 3. Acontroller according to claim 1, wherein each of said basic controlmeans includes:a subtracter for producing a difference between athickness value detected by the thickness gauge in a predeterminedposition along the width of the film and a set value of thickness in thepredetermined position, an integrator for time-integrating thedifference of thickness produced by said subtracter, a memory forstoring past time sequence data of operation amounts of the controlmechanism during a time equal to a sum of the dead time L₁ and a time L₂until the thickness gauge reaches an end of the film after detection ofthickness in the predetermined position, an operational calculator forproducing the past time sequence data of operation amounts of thecontrol mechanism stored in said memory and an estimated value of statevariable at a time earlier than a time when the set value of thedetected thickness value of a film has been inputted by a dead time L, astate shifter for receiving an output of said integrator and an outputof said operational calculator and multiplying a coefficient forshifting the state by the dead time L to produce a state estimated valueat a predetermined time, a state prediction device for receiving thepast time sequence data of operation amounts of the control mechanismstored in said memory to produce state variation based on establishmentof input from a certain time to a time after the lapse of the dead timeL, an adder for adding an output of said state shifter and an output ofsaid state prediction device to produce the state estimated value at thepredetermined time, and an operation amount commander for multiplying astate estimated value at a certain time produced from said adder by astate feedback gain to produce an operation amount command value for thecontrol mechanism(1) a detector for detecting, from said thicknessgauge, a value y(k+1) of film thickness composed of y₁ (k+1), y₂ (k+1),y₃ (k+1), y₄ (k+1) and y₅ (k+1), at a calculation execution timet=t_(k+1) of a time interval T for each calculation execution timet=t_(k+1) each time the thickness gauge reaches an edge of the film andfor producing an end identification signal d which indicates the endwhich the thickness gauge has reached; (2) said detector being arrangedfor supplying the value y₃ (k+1) of the detected film thickness valuey(k+1) to said subtracter for producing thickness deviation (k+1)=r₃(k+1)-y₃ (k+1) between the detected value y₃ (k+1) and a set value ofthickness r₃ (k+1); (3) said subtractor being arranged for supplying theintegrator 102 is supplied with the thickness deviation (k+1) from thesubtractor 101 and producing a time-integrated value of the thicknessdeviation from the following equation;

    X.sub.I (k+1)=x.sub.I (k)+0.5(t.sub.K+1 -t.sub.K){(k)+(k+1)}(40)

where ε(k) is thickness deviation at the last thickness detection time(t=t_(k)) and X₁ (k) is an output of the integrator 102 at t=t_(k) ;said control mechanism including a heater, the integrator includes anexternal disturbance compensator to compensate external heat varying thethickness y with heat generated by the heater so that the thickness y isalways maintained to be a set value;(4) said operational calculatorbeing arranged to calculate, when the thickness gauge reaches either endof the film and the thickness gauge produces the arrival endidentification signal d, a value ω(k+1) from the past time sequence dataof heat generated by the heater stored in said memory and produce anestimated value X(t_(k+1) -L)=ω(k+1) of the state variable at timet(_(k+1) -L) earlier than time t_(k+1) by the dead time L determined bythe arrival end identification signal d produced by the thickness gauge;(5) said state shifter being arranged to multiply the state estimatedvalue [X_(I) (k+1), ω(k+1)]^(T) at time (t_(k+1) -L) by a coefficient e^(L) for shifting the state by the dead time L to obtain the stateestimated value e ^(L) [X_(I) (k+1), ω(k+1)^(T) ] at time t_(k+1) inresponse to the output X_(I) (k+1) of the integrator and the outputω(k+1) of the operational calculator determined by the arrival endidentification signal d of the thickness gauge to obtain the stateestimated value at time t_(k+1) ; wherein the magnitude of the dead timeL depends on the end of the film which the thickness gauge reaches andthe coefficient e ^(L) is different depends on the arrival endidentification signal d of the thickness gauge; said state predictingdevice being arranged to respond to the state shift for the dead time Lto produce a value I(k+1)(6) said memory being arranged to store anamount of shift of states in the form of time sequence input dataexpressed as u(k-2), u(k-1) and u(k) applied to the time domain fromtime t(_(k+1) -L) to time t_(k+1) ; said state predicting device beingarranged to calculate I(k+1) depending on the arrival end identificationsignal produced by the thickness gauge; the past time-sequential datau(K-2), u(k-1) and u(k) being generated by the heater and determined bythe magnitude of the dead time L stored in the memory and being suppliedto the state prediction device from time (t_(k+1) -L) to time t_(k+1);(7) said adder being aranged for adding the output e ^(L) [X_(I) (k+1),ω(k+1)]^(T) of the state shifter 105 and output I(k+1) of the stateprediction device to produce the state estimated value [X_(I) (k+1),X(k+1)]^(T) at time t_(k+1) ; (8) operating amount commander beingarranged to generate an amount u(k+1) of heat generated by the heaterfrom time t_(k+1) to next calculation time t_(k+2) is defined by thefollowing equation using state feedback gain (f₁, F₂);

    u(k+1)=-f.sub.1 X.sub.1 (k+1)-F.sub.2 X(k+1)               (41)

the adder being arranged to supply the state estimated value [X(k+1),X(k+1)^(T) ] at time t_(k+1) to said operation amount commander 108 forheat generated by the heater; said operation amount commander beingarranged to multiply the state estimated value [X(k+1), X(k+1)]^(T) bythe state feedback gain to determine a command value of heat generatedby the heater; and (9) the above control calculation is executed afterthe next detected value y(k+2) of film thickness is obtained from thesampler 100 at time t=t_(k+2) of calculation execution when thethickness gauge is moved along the width of the film after the timeperiod T and reaches the opposite film end.
 4. A controller according toclaim 1, wherein each of said basic control means includes:a subtracterfor producing a difference between a thickness value detected by thethickness gauge in a predetermined position along the width of the filmand a set value of thickness in the predetermined position, anintegrator for time-integrating the difference of thickness produced bysaid subtracter, a memory for storing past time sequence data ofoperation amounts of the control mechanism during a time equal to a sumof the dead time L₁ and a time L₂ until the thickness gauge reaches anend of the film after detection of thickness in the predeterminedposition, an operational calculator for producing the past time sequencedata of operation amounts of the control mechanism stored in said memoryand an estimated value of state variable at a time earlier than a timewhen the set value of the detected thickness value of a film has beeninputted by a dead time L, a state shifter for receiving an output ofsaid integrator and an output of said operational calculator andmultiplying a coefficient for shifting the state by the dead time L toproduce a state estimated value at a predetermined time, a stateprediction device for receiving the past time sequence data of operationamounts of the control mechanism stored in said memory to produce statevariation based on establishment of input from a certain time to a timeafter the lapse of the dead time L, an adder for adding an output ofsaid state shifter and an output of said state prediction device toproduce the state estimated value at the predetermined time, and anoperation amount commander for multiplying a state estimated value at acertain time produced from said adder by a state feedback gain toproduce an operation amount command value for the control mechanism(1)the thickness gauge constitutes means for producing the detected valuey(k+1) of film thickness (vector consisting of y₁ (k+1), y₂ (k+1), y₃(k+1), y₄ (k+1) and y₅ (k+1) at the calculation execution time t=t_(k+1)of the time interval T each time the thickness gauge reaches an end B orC of the film width and produces the arrival end identification signal dwhich indicates the end which the gauge has reached; (2) said subtractorconsitutes means to receive a value y₃ (k+1) of the detected filmthickness value y(k+1) and produces thickness deviation ε(k+1)=r₃(k+1)-y₃ (k+1) between the detected value y₃ (k+1) and a set value of athickness r₃ (k+1); (3) the integrator constitutes means for receivingthe thickness deviation Δ(k+1) from the subtracter and producing atime-integrated value of the thickness deviation from the followingequation;

    X.sub.I (k+1)=XC.sub.I (k)+0.5(t.sub.k+1 -t.sub.k){ε(k)+ε(k+1)}

where ε(k) is thickness deviation at the last thickness detection time(t+t_(k)) and X_(I) (k) is an output of the integrator 102 at t=t_(k) ;said control mechanism includes a heater; the integrator includes meansfor forming a function of an external disturbance compensator and servesto compensate external heat varying the thickness y₃ with heat generatedby the heater so that the thickness y₃ is always maintained at a setvalue;(4) the thickness gauge constitutes means to produce an arrivalend identification signal d in response to the thickness gauge reachingeither end of the film the operational calculator constitutes means torespond to the identification signal d and to the past time sequencedata u(k-2) and u(k-1) of heat generated by the heater stored in saidmemory together with the film thickness value y(k+1), said operationalcalculator consituting means for producing an estimated value X(t_(k+1)-L)=ω(k+1) of the state variable at time t(_(k+1) -L) earlier than timet_(k+1) by the dead time L determined by the arrival end identificationsignal d produced by the thickness gauge;(5) said operational calculatorconsituting means for calculating the state estimated value [X_(I)(k+1), ω(k+1)]^(T) at time (t_(kjl) -L) by a coefficient e ^(L) orshifting the state by an average dead time L to obtain the stateestimated value e ^(L) [K_(I) (k+1), ω(k+1)]^(T) at time t_(k+1) ; saidstate shifter constituting means for multiplying the output X_(I) (k+1)of the integrator and the output ω(k+1) of the operational calculator bythe coefficient for shifting the state by the average dead time L forboth end of the film to obtain the state estimated value at time t_(k+1); said state prediction device consituting means for correcting thestate shift of the input u(k) applied in time domain for only theaverage dead time L;(6) wherein the term I(k+1) represent the shift ofstates for time sequence input data u(k-1) and u(k) applied to the timedomain of the average dead time L from time (t_(k+1) -L) to time t_(k+1)said memory consituting means for supplying the past time sequence dataof the heat generated by the heater determined by the magnitude of thedead time L to the state prediction device from time (t_(k+1) -L) totime t_(k+1) ; (7) said adder further constituting means for adding theoutput e ^(L) [K_(I) (k+1), ω(k+1)]^(T) of the state shifter and outputI(k+1) of the state prediction device to produce the state estimatedvalue [X_(I) (k+1), X(k+1)]^(T) at time t_(k+1) ; said state shifterfurther constituting means for integrating the state estimated value attime t_(k+1) ; (8) an amount u(k+1) of heat generated by the heater fromtime t_(k+1) to next calculation time t_(k+2) is defined by thefollowing equation using state feedback gain (f₁, F₂);

    u(k+1)=-f.sub.1 X.sub.1 (k+1)-F.sub.2 X(k+1)

said operation amount commander constituting means responsive to thestate estimated value [X(k+1), X(k+1)]^(T) at time t_(k+1) from theadder, of heat generated by the heater, for multiplying the stateestimated value [X(k+1), X(k+1))]^(T) by the state feedback gain todefine a command value of heat generated by the heater after obtainingthe next detected value y(k+2) of film thickness from the sampler attime t=t_(k+2) when the thickness gauge is moved along the width of thefilm after the time period T and reaches the opposite film end.